Native English speakers generally use 'Cover' in a sentence while referring to paying a bill or supplementing somebody else's part of the bill.
To be honest, as I'm so used to saying it, I'm not 100% sure if it is 'correct' to use but it is definitely not incorrect.
An example of me using 'Cover' in a sentence:
Can you cover my part of the bill? I will pay you back as soon as I get the money.
Don't be afraid to use it in a sentence as it makes perfect sense and everyone will understand what you're trying to say.
In a nutshell, the act of straining separates solid from liquid, regardless of whether we want to keep either, both or neither.
The object of "strain" (without "off") is the combination of the solid and liquid.
Most often, we refer to the combination of solid and liquid with just the name of the solid because the liquid is not considered valuable enough to mention. Like when we "strain pasta", there's also water in the pot, but since it's worthless once the cooking is done, we refer to the pot containing both pasta and water as simply "the pasta". It's in this sense that we "strain the pasta", which really means "strain the combination of the pasta and the water".
The meaning of "strain off" is the same as "strain", but it focuses on the liquid to be removed. The object of "strain off" can be either the combination of the solid and the liquid, or just the liquid.
So, to your examples, 1, 2 and 3 are definitely correct, and maybe 4 as well. Sentence 2 and 3 you've correctly figured out already. 1 is also correct because in that context, "vegetables" refers to "the combination of vegetables and liquid", which is a valid object of "strain", and we understand from the context that there's water to be strained off.
In sentence 4, if the water is valuable (odd, but possible), then it means to remove the solid from the water.
Also, if there's vegetables cooking, someone could point at them and say, "strain the water". Technically this isn't correct, but it would be clearly understood to mean "strain off the water", and might not be considered a mistake.
Best Answer
I don't know if dictionaries will make the relatively subtle distinction clear, but...
...whereas...
Obviously in many contexts there's actually no difference between refraining from doing something (which you may or may not already be doing when you're asked to "hold off"), and waiting (until the speaker does something).
If in doubt, though, you should probably use on, since it works more naturally in a wider range of contexts (and it might be considered rude in some contexts to tell someone to stop doing something).
Note that a few centuries ago, Hold! could be used with no preposition as an imperative meaning [You] Hold on! / Stay back! a usage which has long since died out. But there is a current prepositionless usage whereby someone on the telephone says Please hold, meaning Please wait [while I put you "on hold"] until I can respond appropriately to your call.