No. We are talking here about 5-13 years old kids. The concepts that the source talks about are very sophisticated. When we are talking about little children, a level of simplification is needed. I studied (now finishing) a double degree in Physics, and Electrical Eng. and what I've seen is that even in undergraduate and graduate courses in the university, courses that are intended to Electrical or Software Engineers often simplify the physics involved ,to a degree that wouldn't be in physics courses, for better understanding of the general more important principal. For example, you don't need to learn quantum physics, only to know that an electron can only be in discrete levels in an atom.
I'll touch on the 8 claims that you provide here.
Some of the claims that the source makes are plain false. For example, the first and the "static charge" statements. Other statements that the source "falsifies" are right inside a solid materials, and others are right inside metal, since semiconductors, plasma, and Ion currents are quite complicated materials, and that children don't often get to see plasma streams or semiconductors, but most of them associate electricity only with electric wires and maybe simple electric instruments, like light bulbs switches and ovens it's safe to assume that when you talk about electricity to children, even if you don't state so explicitly, you are assuming current inside metal conductors.
Lets look over the statements provided here, and see if they are right. My sources is what I was taught during my physics Bsc. For references you can see the book that was the course book when I studies the basics of Electricity and Magnetism, Electricity and Magnetism from Berkley, or you can watch the course filmed at MIT by Professor Walter Lewin. Some of the answer include some complex material from more advanced courses or subjects, at the end I'll give a list of the books which were the course books when I took those subjects.
All electric currents are flows of electrons?
Yes, they are. There is the concept of positive charge flow, where the charge carriers are holes, however, at the micro level, it is still a flow of electrons, on the opposite side that creates a lack of electrons which "moves". However this is NOT intro level physics, but taught at solid state courses or semiconductors courses. Just to clarify, holes are positive charge carriers, however, holes are not a real particle, they are a representation of a lack of electrons.
EDIT: after consulting a physics professor and doing some more research into this, I realize I was wrong here. Here is the fixed version:
Protons are ionized hydrogen atoms, Ion conductance is present in Plasma, liquids, and can be present in some solids. But this isn't present in metals and wires, or any "classic" electrical appliance. since we assume that we are talking to a 13 years old kid, we don't talk about plasma, neurotransmitters, or semiconductors, so yes, inside a metal wire all electric current is due to the displacement of electrons.*
"Electricity" is made of electrons, not protons? Nope.
Charges of "electricity" are carried both by electrons and protons
Not sure what "Electricity is made of electrons, not protons?" means, because Electricity is a concept, not a real things, just like saying that gravity is made of mass, and relativity is made of light.
If they mean that both protons and electrons can form electric current, then see previous answer.
Electrons are a kind of energy particle? Wrong.
Electrons and protons are matter, not energy.
Electrons and protons are matter and not energy, photons are energy particles. However, when we talk to a 13 years old kid about transferring energy from the power station to their home, then saying that the powerstation is providing energy in the form of a current that moves electrons, it's a completely reasonable simplification.
"Electricity" carries zero mass because electrons have little mass? No.
Quantities of "Electricity" (meaning charge) have weight because charge is part of matter particles
The only thing I have to say is WTF?!, electrons have mass, and so do protons, but charge is a property, it doesn't "carry mass" it's like saying that because things have length and color, then "length carries a color". There are 2 types of particles that have charge: leptons (electron, muon and tau) and quarks, each of those particles has mass and charge, but charge and mass are properties of the particle, not a particle by itself (protons and neutrons are made of quarks). This is particle physics material.
If the meaning is that electrons have no mass, then yes, they don't have mass. Electrons have ~10^3 less mass than a proton, for all practical purposes, an electron doesn't contribute any mass to the mass of the atom. Since, again, we are talking with kids, and we don't calculate the mobility of different materials (Solid state physics), or about movement of individual electrons (Quantum Physics), then the approximation that electrons have no mass is good enough.
Positive charge is really just a loss of electrons? Wrong.
Positive charge is not made of "missing electrons." Positive charge is a genuine type of charge in its own right.
True, a proton has a positive charge by itself. However since most subjects in nature come with a neutral charge, then in order to have a positive one you will need to take a way a negative charge (electron) and get an Ionized atom with a positive charge. Since kids are taught what is in nature, and nature doesn't have free positive charge without taking away an electron, again, acceptable.
Positive charge cannot flow? Totally wrong.
Electric currents in a metal wire are flows of electrons, but in many other materials both the positive and negative charges can flow
Positive charged particles, i.e. protons can't flow inside a solid matter, only electrons can, in some instances, we will say that there are "holes" that hold a positive charge and they are flowing inside a material, but holes are but a real particle, only a representation for a lack of electrons, protons don't move from their place inside a solid (they can vibrate in place). See first point.
To create "static" charge, we transfer the electrons? Not always.
"Static" or imbalanced charges can be created by removing electrons from a neutral atom.
On the first sentence they say "To create "static" charge, we transfer the electrons" but then they say "created by removing electrons from a neutral atom", which is the same as transferring an electron. You will get static charge if you just "remove" the electron, but the electron has to go somewhere.
THE "ELECTRICITY" INSIDE OF WIRES MOVES AT THE SPEED OF LIGHT? Wrong.
In metals, electric current is a flow of electrons. Many books claim that these electrons flow at the speed of light. This is incorrect. Electrons in an electric current actually flow quite slowly, at speeds on the order of centimeters per minute.
Electricity doesn't move at the speed of light, only light (photons) moves at the speed of light, and the speed of light changes in different materials. This is the only point where I agree with the source. While electrons move at very high speeds, the "current" is their drift speed, which is much slower (I don't know if they are right as to the speed is several cm per sec, but it changes with the magnitude of the applied field, so obviously, the current speed will change, whether you apply 1.5V of 1e5V on the conductor). If there is a text book saying that electrons move at speeds close to the speed of light without saying that the current doesn't flow at that speed, than that textbook is wrong.
Explanation on drift speed vs. speed:
Electrons move in random directions in very high speeds. When we talk about the current speed, we are talking about drift speed. The electron moves in different random directions inside the material, then collides with another particle, loses all it's speed and starts moving in a new random direction. When an electric field is applied, then the electron will move slightly more to a single direction, and thus drift in that direction. This is also why the current speed and not acceleration is relative to the force applied. A good video explaining this subject
What we have here is a post (and I would even go so far as call it a rant) that kids smaller than 14 aren't taught university undergraduate and graduate level physics, with all the fine nuances. While in some cases the source may have merit and a viable point if it was addressing that non-physics students (like EE, CS, MEC and others) are taught over simplified physics, the source doesn't address them, but, instead addresses small children in elementary school. Even if to claim that this is addressed to grown people who don't work at physics related jobs, or have physics related occupations would be, it would still be way too petty, as people who don't need the exact details, could be left very well with a basic understanding of the subject.
You can't explain physics at the level that the source demands without a lot of time being spent at explaining and learning all the subjects that are needed for a full and real understanding, including high level math, basic (university basic = high-school advanced) chemistry, and the physics that is at the core of those laws, including, particle physics, relativity theory, quantum mechanics, statistical physics, thermodynamics, EM theory and maybe even more, go to any college or university web site, look at the list of courses needed to be taken by students before they take the solid state physics course, this all you need to know to understand, why certain particles can conduct in certain material and other in other materials.
Because a full explanation is a very very very long and hard explanation, it's better to leave people and especially kids with some basic knowledge of the principals behind the phenomenons that they see every day, than to leave then with no explanation at all.
You can see from the source material at the undergraduate level that I provide here, that all those nuances and fine details are taught to students that learn the subject at the university, and that there isn't a fear of a physicist who thinks that protons don't have a charge associated with them.
Sources for other subjects touched on here:
Particle physics: No course book, but you can see a full Stanford course on this subject on You Tube
Semiconductors and Solid State Physics: Solid State Physics /Ashcroft & Mermin
*- Electromagnetic theory also knows to deal with magnetic currents, i.e. the displacement of magnetic charge. But since magnetic charge carriers were never found in nature, this is a purely theoretical aspect of the EM theory.