Is electricity really the flow of electrons or is it more involved?
The idea that electricity "does not exist" is just verbal sophistry along the same lines as "matter does not exist, it is frozen energy" or, "you do not exist, you are a figment of your own imagination". At best these are all just over-dramatic and misleading ways of saying that what these things actually are is not what you probably think they are. At worst, misguided eccentrics create "straw" definitions of such well-known words just so they can burn them and trump them with their own untenable notions.
This last, sadly, is what is happening with the pages you link to. Although basically sound at an experimental and phenomenological level (and that has to be clearly acknowledged), he argues for his own wacky definitions of words.
The guy claims that the scientific definition "means only one thing: quantities of electricity are measured in Coulombs". In fact it is electric charge which is measured in coulombs, not "electricity" per se; his claim is a classic example of giving a straw definition so that he can debunk it.
He concludes that "Because there are *two* things flowing, we cannot call them both by the name 'electricity.'" This is typical eccentric pedantry; of course we can. We can simply say that the phenomenon of electricity comprises electric charges and electric fields. And we do.
For example "static electricity" is a buildup of charge, which creates an electric field capable of making your hair stand up or making molecules of glue stick together. Electricity as found in domestic wiring has three main properties; its flow is measured in (electron) charges per second or Amps, its "pressure" in Volts and the power (flow of energy) carried by the charged electrons in Joules per second or Watts. All these are aspects of electricity. There are many others.
All the sophistry in the world will not change these facts, only what we choose to call such things.
Electricity is not a well-defined term in physics. It's a layman's term that means something like what physicists call electrical phenomena. However it also gets used (like your first link says) for numerous specific phenomena that physicists have more specific terms for:
So if you're talking to your cousin you can tell them "the electricity's out" or whatever. But when you're discussing physics you'll express yourself more clearly if you use the specific terms for the specific aspect of electricity that you want to discuss.
Electric current is the flow of charges.
In most metals, electrons are the majority charge carrier, but that's not the case for all materials. For example, in pure water, there aren't free electrons, and the charge carriers are H$^+$ and OH$^-$ ions. There are not many ions in pure water, so it's a poor conductor. But dissolve some table salt and you introduce Na$^+$ and Cl$^-$ ions and improve the conductivity a great deal. A chemist could tell you which is the majority carrier, as well as a bunch of other stuff that's happening.
In some materials, especially p-type semiconductors, the most parsimonious description of the majority charge carrier is that they are positively-charged "holes" in the electron ocean filling the crystal. Whether those quasiparticles are "real" in the sense that electrons are real is one of those questions that gets more slippery the more you think about it.
You link to Bill Beaty's amasci.com. The more I learn about electromagnetism, the more Beaty's style and content impress me. I say you keep reading.