It's a bit more narrow than this. Jokes, sarcasm, all kind of cheerful additions are almost always acceptable and welcomed unless you actually discuss some actual tragedy. "Foreigners" just don't joke in the same way, as it's hard to deliver a punch line, not to mention a properly timed one. I think that "personality" you talk about is still there on both sides, but is very different. The "native" one is great but hard to understand.

But English speakers that I talked with (or listened to online) love actual wordplay. It's part of their culture. Tiny rhymes, replacing words with wrong ones that sound similar and "everyone knows what he meant".

Anything that you could comment as "pun intended" or "pun not intended" might be super hard to grasp for a foreigner. It often requires listener to know correct pronunciation and some phrasal verbs to get it and often knowing some facts from that country. I usually understand these things when said by Americans, but lack knowledge to understand similar "puns" by other nations. I know for a fact that most of my colleagues don't understand most of those. They might get that there was a joke there, but they won't know if it's commenting the topic positively or negatively.