When something becomes sunken, we can say it is depressed. To cause this state is to depress it. What is the opposite? Undepress isn't in the dictionary.
The context is "When we click the button a second time, we _____ it."
EDIT: I'm referring to the observable behavior/appearance of the button. Describing the user's behavior is not useful here.
"When we click the button a second time, it _____."
I would use the word "unclick".
Especially when applied to graphical interfaces, unclick is widely understood in the Internet Age as a synonym for "deselect" or "un-choose". When specifically referring to check-box elements, uncheck is also used.
The confusing part here is that you are using "press" to "depress" and then also using "press" to "undo" the cause of the same action. You then switch to the sound a button makes ("click") and use that to refer the the "un-pressing" action. All of which is acceptable and illustrate why this is not straightforward.
"Toggle" means to "switch between states" so a "press" could perform the action of "returning to the previous state" regardless of the present state. So you could say "depress" and then "when we click/press the button a second time, we toggle it back."
In our shop we refer to the states as "pressed" and "unpressed" (which, oddly, is defined but not for this use, but we use them anyway). Then frequently refer to actions on the button as a "toggle" ("click to toggle" or "press to toggle") especially in cases where we do not want to or cannot accurately specify it's current state.
In some programming languages or scripting languages such as AS, VB, JS button states could be Up / Over / Down OR pressed / unpressed
for html regular form button, there is no such state something "pressed" or "unpressed". When you click a button it goes to "down" and comes back automatically.
Anyway, you may click a button second time to "toggle an action" - not button itself
"Bulges out" comes to mind, but would be a little out of context for a graphical button. "Toggle" describes the action of clicking the button. But if what you are trying to describe is the physical appearance of the button after the second click, I would go with "restore" or maybe "enable".
I think "..., it releases." Clicking it the first time leads to it being "caught" in a lower position. Clicking it the second time releases it so that it can go back to the original position.
When we click the button the first time, we depress it (suggested by the OP)
When we click a second time, we pop the button up
Alternatively, we need to slightly rephrase the instruction.
Clicking a second time will pop up the button
to pop (intransitive verb) 1 a. to go, come, or appear suddenly —often used with up
Merriam-Webster
NB: A pop-up (with the hyphen) is either a noun or an adjective and often refers to a window that appears suddenly on a computer screen. For example: many unsolicited ads are pop-ups; clicking here will produce a pop-up menu.
