The report referenced by PETA was updated in June 2007 by the American Veterinary Medical Association, and is now called: AVMA Guidelines on Euthanasia
They do discuss electrocution techniques, with references.
Electrocution, using alternating current, has been
used as a method of euthanasia for species such as dogs,
cattle, sheep, swine, foxes, and mink.
Electrocution induces death by cardiac fibrillation,
which causes cerebral hypoxia.
However, animals do not lose consciousness for 10 to 30 seconds or
more after onset of cardiac fibrillation. It is imperative
that animals be unconscious before being electrocuted.
So, yes, electrocution takes time - it doesn't cause immediate death - but there are techniques (including "electrical stunning") which avoid the animal suffering.
Electrocution is humane if the
animal is first rendered unconscious.
If the PETA depiction is correct, this requirement is not being met.
AVMA warn it is not suitable for all animals:
It may not result in death in small animals
(< 5 kg) because ventricular fibrillation and circulatory
collapse do not always persist after cessation of current
AVMA do not explicitly mention genital or anal electrocution, but they do say:
Techniques that apply electric current from
head to tail, head to foot, or head to moistened metal
plates on which the animal is standing are unacceptable.
Presumably, "head to tail" covers the scenario described by PETA, and is therefore considered unacceptable.
In summary, PETA are correct that electrocution without first being rendered unconscious does lead to a slow death. They are correct in saying such techniques are considered unacceptable by the AVMA.
I have not attempted to address any of the other claims that PETA makes - e.g. that fur farms are actually using such techniques and are doing so without first rendering the animal unconscious, and that no laws about the humane treatment of animals protect them.