As far as the less-clear cases, this is basically CAP theorem, with some physics thrown in for good measure. In some sense there is always a partition of some length between two points, thanks to the speed of light: the theoretical limit of information propagation through space.
So in the presence of this delay "partition," you have three choices, really, and the choice you make depends on the application.
A) You can choose to be available and responsive. Show the user feedback immediately and never concern yourself with global state. Technically, I'd call this an illegal choice because you must have some sort of state to even be executing code. Unless you simply don't write the code, in which case your job is easy!
B) You can choose to be immediately available and eventually consistent. You calculate the response quickly with the assumptions you have most available (local memory, disk), all while transmitting events and waiting for the further-away less-available state to become available.
This is the way many online games that need quick feedback to be fun are done. [1] Unfortunately, this is also the source of the lag jumps that you see. You're always running with [partition-size in ms] outdated global state, so the assumptions you made when calculating outcomes are going to be incorrect. This is why your headshot might turn into a total miss when the player jumps five feet to his right and, oh yeah, you also died.
3) Don't react to events until the global state has been updated.
This means a full round-trip plus processing remotely and locally before that click event performs the action it is supposed to. This can be anything from a crappy experience (I shot into the ground, why should I wait), all the way to the only sensible choice (if integrity is highly important, say in transactions and avoiding double-spend).
Really, it's so much more than this too. On top of availability vs consistency you have to account for some trust model (the game client says it was a headshot, but how do I know I can trust the client) and information security (confidentiality, availability, integrity).
So TL;DR there are lots of very hard problems in distributed systems and sometimes people just default to one stance or the other to balance their cognitive load or for any number of reasons (ranging from legit to ridiculous). Sometimes they default to consistency. That's probably the case for your button-click example.
[1] https://developer.valvesoftware.com/wiki/Source_Multiplayer_...