Ability checks are intentional, Saving throws are reflexive.
An Ability check is...
when a character or monster attempts an action (other than an attack) that has a chance of failure.
A Saving Throw...
represents an attempt to resist a spell, a trap, a poison, a disease, or a similar threat. You don’t normally decide to make a saving throw; you are forced to make one because your character or monster is at risk of harm.
And there you can see that difference rather plainly. An ability check is an intentional act--something a character voluntarily elects to do in order to try and accomplish something. A Saving Throw is a reflexive response to something trying to do your character harm.
So, in your specific examples...
In the first, the fighter is intentionally choosing to try to prop up the ceiling. This will be an Ability Check. In the second, the rope bridge abruptly shakes and the characters are forced to react to not fall off. This will be a Saving Throw.
To clarify some things mentioned in the comments...
Grapple checks are an Opposed Check. Resisting a grapple is not something that is a reflexive action. One may flinch when someone goes to grab you, but out-muscling them (Athletics) or twisting free of their grab (Acrobatics) is something you have to do on purpose. (Speaking from experience as a Martial Artist, here). Thus, it is still an ability check. It is an intentional thing that a character chooses to do in order to accomplish something.
Perception, likewise, is something a character elects to do. They are actively looking for hidden threats, looking for hidden doors, etc. And, in combat, a Perception Check is treated as an Action. Passive Perception represents your character's ongoing intentional attempts to look for trouble.
Again, in both of these cases, the character is intentionally choosing to do these things. Hence, they are Ability Checks.