When quit, I am without a job but the boss still has his. When I am fired, the same holds except that I didn't even choose when it would happen. They are not symmetrical, as I can never choose to stay while making the other person leave. At best, I can choose /when/ I leave.

But also, as a practical matter, most "bosses" (line managers at least, not sure how true this is further up the org chart) in IT can't actually fire their reports. /They/ would be fired /themselves/ if they did. Instead, they are expected to "manage out" unfavored employees by just giving a few quarters of bad reviews, performance plans, write-ups, etc. until they start interviewing elsewhere and leave on their own.

Of course there are power dynamics at play here - more or less of them depending on how crappy vs decent the company is.

However, I don't see any power play in the words themselves. An employee being fired and an employee resigning are different concepts, and different concepts get different words.

You could read something into the word "firing". But don't most companies avoid that word in any case, in favor of "letting go" or "dismissal"?

It is an interesting thought to play around with - but probably not. If a company is allowed to frame firings they would use mild language and try not to refer to the situation at all ('the business was restructured', 'redundancy', 'layoff'). If the firing was for-cause the language gets a bit stronger ('terminated' for example would indicate that the person had done something that scared a team of lawyers or was otherwise egregious).

Firing is a slang term that would tend to be used by an unsympathetic press. The slang equivalent would be quit which has a similar strength.