In post #13 of this thread you said, "Displacement current is a valid form of current and it is the current the exists in the capacitor dielectric." Since "displacement current" is an equivalency and not a real physical current, and doesn't exist in the dielectric, I can be forgiven for assuming you are confusing the real current in a wire with the assumed current in the capacitor.
I did say that, which is not the same thing as saying displacement current is drift current. But, I see that what I said is not quite right, so I need to correct it. The displacement current is really better described as something due to fields in space, rather than in the dielectric. The dielectric is better said to have polarization current, which is what gives the dielectric its higher effective permittivity.
As I said above, even the drift current does not go through the wire with AC, the charges on average move back and forth. At 60 Hz, the drift can only be about 10 μm in either direction, and 60 Hz is a relatively low frequency. That's hardy describable as "going through" but we say it anyway because that's the terminology that exists. In the same way polarization current does not represent charges actually going through the dielectric, but we say it anyway, and we know what it really means. It means charge going back and forth, but in this case the charge is bound to an atom and the electron cloud distorts. With polarization current the charge movement is in Angstroms rather than μm.
Is language always perfectly descriptive? No, of course not, yet it communicates effectively when we rely on the common knowledge base rather than a literal interpretation of words.
The author says in the 5th paragraph "The current through the capacitor (red) will 'lead' the mains voltage (green) by 90º. " Since he says the current passes through the capacitor, he is also saying that current is passing through the dielectric, which it does not. He should have said something to the effect of (the branch containing the capacitor). Drift current is the only type current in a wire, so the author can only mean drift current.
No, this is your own interpretation. I would interpret what he said differently. You seems to like to enforce your own definitions and try to impose them on the rest of the world. The term current is general and can include drift current, relative charge movement of any type, polarization current and displacement current. That's just terminology. It seems they are all grouped together for good reason, since one gives rise to the others and all are consistent under Maxwell's generalized version of Ampere's law which includes displacement current. Yes, there are some people that don't like these definitions, but too bad, because they are already defined and existing in the history and literature. You are free to include the adjective if you want to say "drift current" but don't require that the rest of the world should mean "drift current" when they say "current". We all understand the difference (or at least we should) between the types of current that have been classified, and when someone talks about current going through a wire and through a capacitor, we understand (or at least should understand) what is happening physically, and we should not need to every time write several sentences of clarification to say such a well understood thing.