Coming back to this a couple days later, my intuition (native English speaker) is:
climb in the tree: start aboveground, finish aboveground
climb (up) the tree: start at ground level, finish at the top of the tree
"On" and "onto" both aren't words I'd see used often with "tree," but I've gotten in this habit of translating the Hungarian sentences word-for-word, especially in the prefixes and case-endings units, because it's seemed like the best strategy for having my answer marked correct.
I guess my original comment wasn't well-thought-out, and what I was really wanting was some more consistency: accepting the naive word-for-word translations either (almost) everywhere, or (almost) nowhere