"Repeated wash and rinse is required for most people?" – No – no need!
Earliest usage or origin?
Rumour has it that David Ogilvy invented this "trick" for Proctor & Gamble". But it is unlikely that he invented this strategy, as should be obvious from Edna Kent Forbes' "Beauty Chat" (The Border Cities Star - Aug 8, 1927):
Wet the head all over, drip on some soap syrup, lather, rinse. Repeat,
rinse, repeat a third time and rinse. Let this last rinse be fairly
warm and squeeze into the water the juice of one good sized lemon.
So, at least this Ogilvy/P&G theory doesn't hold much water. One connection seems to come from instructions for dying wool, preferably black, and finally washing wool.
In Britain the newspapers gave "mutual aid advice" for cleaning wool and oilskins in this way since the mid 19th century. As of 1908 the New York Times emphasises one lathering but four rinses. And starting in the nineteen twenties there are ads for Palmolive soap ads utilising the procedure frm he claim:
Palmolive Soap.  at the regular price. Repeat both washing and rinsing. Use cold cream you wish, *' exceptional offer open but only you have natural March 31st, and ... (1925)
But for the actual claim about the effectivness: There seems to be conflicting evidence out there:
"Expert" opinions on effectiveness
One Simple Trick that will Transform your hair, According to a top Hair Stylist:
According to top hairstylist Ian Florey, we’re washing our hair all wrong - it’s critical to wash it twice, he says.
The club for men with long hair called The Longhairs
Whilst many people presume the ‘lather, rinse, repeat’ instructions on shampoos is just a way to get you to use more of the product, Florey waxes lyrical about the benefits of doing so.
“Wet the hair, then wash it twice with a non-sulphate shampoo. The first shouldn’t create much lather but the second one will give a lot more,” he explains to The Independent.
Florey says it’s essential to shampoo twice, rinsing in between, and your hair will be transformed if you do.
“The hair will become a lot stronger, shinier and healthier,” he says, but adds that the improvement won’t be instantaneous and it could take a few weeks or even months for the change to take effect.
The reason you need to shampoo twice is that the first wash removes all the dirt (and thanks to pollution, dry shampoo and styling products, there can be an awful lot built up), and so you need to wash it again for any of the nourishing properties of the shampoo to get to work.
“The first shampoo is to cleanse the hair and the second shampoo is to add anything into the hair - be that strength, moisture or protein,” Florey explains.
What’s more, you only need a dollop of shampoo the size of a 20p coin - provided you’re using shampoo of a good enough quality which should lather up easily.
Florey has over thirty years’ experience in the world of hairdressing, has worked on designer fashion shows including Julian McDonald, Erdem and Roksander Ilincic, and now works at the trendy hair salon-cum-art gallery The Lion and the Fox in Clerkenwell, London.
This offers a least a possible explanation for effects, if any.
But it also the point of view of a believer. Contrast that view of a priest with the view of a bishop:
Wash , Rinse, Repeat: Lessons from one of the greatest marketing ploys of all time
We have all seen this statement on our shampoo bottle under “instructions”; Wash, rinse and REPEAT! Well it says it on the instructions so I must need to use twice as much of XYZ’s product to clean my hair. With one word, the XYZ company theoretically doubled its sales by encouraging the consumer to use up the product twice as fast. Brilliant.
Now what can the service industry learn from this simple word; repeat. Repeat is a concept that has plagued businesses for the millennial. How can a company garner the repeat, or three-peat or the four-peat and so on? What makes guests, clients and customer come back to Jane Dry Cleaner, Moe’s Tavern or Curly’s Hotel? What magic word can be used to lure the increasingly fickle, price comparing, experience driven modern day consumer back to an establishment previously visited?
Shampoo bottles frequently display the instructions to wash, rinse, and repeat. This implies that repetition improves results. You should do the same. Deliver on your commitments, then do it gain. Consistency builds trust.
Scanning through women's self help forums reveals a wealth of opinion on this. But no consensus. Doing the same search over articles published in women's magazines gives you a strong consensus. Trying to paraphrase that: "This is a real effect and you should do it and here is an affiliate link to the finest products for this on the market today that you can use."
The above strongly suggests that the shampoos contain something "that works" on your hair or even in your hair. Unfortunately they most often do not. The effect any ingredient in there could have must be weak. If they did display a strong effect they would be classified as medicinal products under some jurisdictions (like in the EU). If the product in question is identical to the one sold in the EU this reasoning applies.1
Examining medicinal shampoos in a pharmacy I did not find the instructions in question on any bottle. Looking at bottles from other (foreign) markets, even those that are sold for dandruff prevention etc, I did not find the instructions in question. Manufacturers showing off these claims as scientifically proven are non existent. Some are even on record flat out denying this "need":
[…] advances in shampoo technology mean that only one application of, for instance, Clairol's Herbal Essences is sufficient to break through the oiliest hair. The company has stricken the use of both REPEAT and REPEAT IF DESIRED from all Clairol products. Yet a lot of brands, like Suave by Unilever and L'Oreal, still say REPEAT. Others, like Unilever's Finesse and Revlon's Flex, opt for the less imperative REPEAT IF DESIRED. Procter & Gamble uses REPEAT IF NECESSARY on Pantene.
Getting consumers to wash twice can, of course, increase sales--in ways one might not imagine. Double sudsing leads to dry hair, Fekkai points out, and that means more beauty products! "When you do two shampoos, even if you don't usually use a conditioner, you have to use a little," he says. "The conditioner becomes very important." REPEAT. FOLLOW WITH CONDITIONER. Words Cheever's marketer could have retired on.
To quote the peculiar stylist again: "The reason you need to shampoo twice is that the first wash removes all the dirt (and thanks to pollution, dry shampoo and styling products, there can be an awful lot built up), and so you need to wash it again[…]" This might be actually a legitimate reason hidden in plain sight: All those products in your hair are very hard to really get out properly again.
Rinse and repeat (Manufacturer link):
‘Lather, rinse, repeat.’ The instructions for shampooing are basically iconic… but not necessarily right. Repeating the shampooing step might be useful if you’ve been hitting the hairspray hard or have naturally more oily hair, but for daily shampooing it’s not necessary
While one thing is certainly true: a first wash removes the dirt and oil. Oil will reduce the foaming. A second round will feel and look different and produce more foam with less product. But since your hair and scalp are already clean the second round is just a waste.
Hard evidence seems to be lacking either way, but all those circumstantial points are a clue that this is indeed more marketing than proven effect. A second wash is certainly not required.
1 Medicinal Products Directive 65/65/EEC (MPD):
“Pharmacological means”, in the context of the MDD and AIMD, is understood as an interaction between the molecules of the substance in question and a cellular constituent, usually referred to as a receptor, which either results in a direct response, or which blocks the response to another agent. Although not a completely reliable criterion, the presence of a dose-response correlation is indicative of a pharmacological effect.
“Immunological means”, in the context of the MDD and AIMD, is understood as an action in or on the body by stimulation and/or mobilisation of cells and/or products involved in a specific immune reaction.
“Metabolic means”, in the context of the MDD and AIMD, is understood as an action which involves an alteration, including stopping, starting or changing the speed of the normal chemical processes participating in, and available for, normal body function.