Yes, @YvesLefol, the simple past, I saved up for months…, would be fine. Indeed, at least one frequent contributor here, FumbleFingers, encourages learners not to even bother trying to use the perfect forms until they’ve become pretty highly proficient in English.

(@fumblefingers, are you listening?)

I am! And I disagree with the pedantic "not considered correct grammar", as you probably expected! In the OP's specific example #2, I have no objection to Present Perfect being used by even the most careful speaker, to add "immediacy" to the fact that the saving up extends from way in the past, to only just yesterday. It's a context-specific way of being "emphatic" (to non-pedants! :)

Just out of interest, would you still say the same if it had started This morning I bought a new laptop... ??? How long ago does the purchase have to be before you accept Present rather than Past Perfect for the preceding process of saving? What if the speaker is just leaving the shop? I just bought a new laptop...

Yes, @FumbleFingers, I would still say the same because (1) the form I bought situates the buying in the past, and (2) the saving was performed during some period that had ended before the buying took place.

Btw @FumbleFingers (and in case the down vote was yours), would you have been happy if I had said instead, “but it’s not standard grammar”? I’m going to make that change anyway so the answer doesn’t come off as prescriptive.

Absolutely! I might interpret "non-standard" different to you (I might think it's just "less common", you might think it's "less correct").

@FumbleFingers, I don’t think I’m all that prescriptive, though my kids might beg to differ…

I just think we need to be careful about "correct / incorrect" in the context of learners. There are just so many contexts where in practice it really doesn't matter which of two or more alternatives you use, but questions like this often virtually force many native speakers to endorse just one version. Even though if they were in casual conversation with another native speaker, they wouldn't think anything of it if they heard one of the alternatives. And imho, that's the level most learners should aspire to reach (not the level that gets marks from pedantic examiners).

It's the same as when a learner asks about the difference between two or more words. People feel compelled to find the tiniest nuance of difference (spurious or not), when so often it's not that important.

I respect your position, @FumbleFingers. It reminds me, though, of the sensation I felt when my prof for 3rd-semester Russian announced, “We’ve actually been lying to you about verbs from the start. There’s this thing in Russian called aspect, and it’s so hairy for English speakers to learn that if we’d introduced it right out of the gate, you probably all would have dropped RUS 101.” I understood their pedagogical choice, but I also resented having been taught “fake” Russian grammar. Maybe that experience was influential in my preference to deal with the English perfect ab initio.

Re difference between two or more words, whether it is or isn’t “that important” is in the eye of the beholder. I gather that your model of English learners is that they just want to be able to express themselves adequately to be clearly understood.

And I’m sure that that model does fit many learners. My model, which likely reflects my projecting my own logophilia on them, is that definitional nuances can be fascinating. And as you certainly know, the shades of meaning and fine distinctions made possible by English’s enormous lexicon is a key factor in what makes the language unique.

...Bizarrely, I become relatively conversationally fluent after spending less than a fortnight in a monastery where the monks weren't supposed to engage in idle chit-chat! Apparently, talking to a foreigner didn't count. Whatever - I see so many questions here that have "unignorable" shortcomings in syntax, vocabulary, or idiomacy, yet they ask "which is correct" about alternatives that all seem fine to me.