~てきた translation

I’m learning about ~てきた and am still a bit confused what it actually means. I’ve come across some example sentences in a couple of textbooks such as the following:

日本語の文法が面白くなってきた。
I’m starting to find Japanese grammar interesting

大分寒くなってきた。
It is getting very cold.

暑くなってきた。
It has gotten hotter.

These three sentences translate てきた in slightly different ways so I’m just a bit confused which is the right way to translate it/think about it. In the first sentence the change seems to have just started. In the second sentence the change seems to be ongoing. In the third sentence it sounds like the change has reached some state of completion. Would really appreciate some clarification on what the deal is and why these could all be valid translations or if they’re actually wrong.

by lucario-95

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