I think the correct answer to "what exactly is [Resh Lakish] saying" is simply that the word "ki" may mean different things in different contexts.
It's definitely only one word in Hebrew, and IMHO there's no single underlying concept or meaning uniting all the terms.
Rather, if you're an Aramaic speaker, and you want to translate/explicate/darshen a pasuk, you will find yourself reaching for different terms for the same word.
To give an example from English, I think this is similar to trying to translate the word "with." (Davar b'shem omro, I learned this from Russell and Norvig.)
- I ate spaghetti with a fork.
- I ate spaghetti with dinner.
- I ate spaghetti with friends.
- I ate spaghetti with meatballs.
- I ate spaghetti with abandon.
Every English speaker knows that there's only one word "with." But if you translate those sentences into other languages, you may find that correct idiomatic language use compels you to use different words for different situations.
I think that there is no single concept or meaning here, because the only possibility for that concept would be "togetherness" or perhaps "unity," but squashing all the above meanings into the concept would (a) unite too many disparate things, and (b) not explain when to use "with" and when to use a different word. I expect some people to disagree with me, but not if they fluently speak languages other than English.
I'd also like to give an example of what Resh Lakish is not saying. In Hebrew, the word "`etz" can be translated into Aramaic as either "a`a" (that's אעא) or "ilana." In English, these would be rendered "wood" and "tree," but note that in English you might also need to use "lumber." In this case, it's still a single word in Hebrew, but there is a concrete understanding that it encompasses trees and things made out of what used to be trees.