Which diode to use on my RC switch debounce circuit?
A simple RC circuit should suffice. You don't need that diode, just leave it away.
The most important part of that circuit is the Schmitt trigger, but it seems that you got that right.
As for the power rating, Spehr Prefhany is right, any standard 1/4W resistor will do.
Use a 1N4148 for the diode, assuming through-hole.
For the resistors, 1/4 W is more than adequate. The power dissipation is 25/82K = 0.3mW if you hold the switch down.
The adapter should be fine if it outputs 5V with no load (most will).
I have used that self same circuit (from that very same source) for some time now. It works well, but I don't use the diode. I also don't use 2W resistors, but little SMD 0603 ones.
Also the component values I use are very very different.
For R1 I use 10KΩ. R2 is 100Ω, and C is 100nF.
I also don't use a specific Schmitt trigger gate since most MCU inputs are either Schmitt trigger anyway, or have enough noise rejection to not matter.
Unless you have incredibly noisy switches (I use tactile buttons with this circuit, which aren't that noisy anyway), you don't really need massive capacitance and large resistors to slow down the charge/discharge.