What exactly is a softlock?

Solution 1:

A softlock is an instance where game remains in a playable state, but progressing further or past a certain point becomes impossible. It's an event that can happen to speedrunners where triggers and events are hit out of order, causing the games script to break completely, thus causing the progression breaker.

Contrast this with a hardlock, which locks up the game to the point where command inputs do nothing, and the game becomes unresponsive.

With the examples you presented:

For example, some games (notably old-school adventure games) allow you to save your game and continue playing after missing your only opportunity to acquire an item or trigger an event that is required for victory.

Defeat does not constitute a softlock -- defeat is just another ending for the game. A bad ending doesn't inhibit your gameplay commands nor does it render the game unplayable, it simply means you lost. However, if the conditions for either a bad or a good ending cannot be achieved whatsoever due to previous actions, then this can be considered a softlock.

I've also heard of saving (or auto-saving) immediately before an impending death cited as an example of a softlock.

This is a player-induced softlock, not a game softlock. The player has put themselves into a situation where they cannot progress further, so yes, this is a softlock. Examples would be saving with 1HP with no healing items before a boss you must defeat in order to proceed.

Yet another example is cases where some or all of the controls become unresponsive, or a critical menu screen becomes inaccessible, while the game continues to run normally otherwise.

If the game is inhibiting players from inputting commands or accessing a critical game function that would otherwise allow the player to progress, this is a hardlock, not a softlock.

And of course there's cases where you simply fall into an inescapable pit, get stuck in a wall, or are otherwise immobilized without dying.

This is a softlock. Normally developers have teleport triggers that reset a player back to a playable position in the world in the event they fall off a pit and such (or just kill the player, forcing them to restart at a previous checkpoint). If the developers didn't put in a failsafe in the event of the situations you describe, and you cannot progress because you entered one of those conditions, then you have been softlocked.