Okay, according to the (monolingual) dictionary that I use, てやる is used to
indicate "doing something with strong will or determination" (my translation)
e.g. 飛び降りてやる！ But this definition seems a little too vague for me; is there
anything that it actually translates to in English?

I've seen some sources claim that it still means "do" but I'm unsure, because
I have come across constructs like やってやる！
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question
て-form definitions subsidiary-verbs
What does てやる mean when it is not used for giving?
「Verb + てやる」 has two (very) different meanings/usages.

  1. To offer to do something in a patronizing or condescending tone as to demand gratitude.

  2. To (proactively) perform an action so as to prove one's ability to others.

OP seems to be talking about #2 above.

My own example sentences:

>   1. 「[心配]{しんぱい}するな。１[年]{ねん}くらいオレが[食]{た}べさせ **てやる** 。」 = "Don't worry. I'll
> feed (or 'take care of') you for a year or so."
>   2. 「[見]{み}てろ！２[年以内]{ねんいない}に[大]{だい}リーガーになっ **てやる** ！」 = "Be watching. I
> will be a major leaguer within two years!"
>
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answer