This does not compute for me. Even if you have weak future time reference,
there are still temporal adverbs like 'tomorrow', 'next year', 'soon' etc.
in your language. And actually, 'tomorrow' was present in the given example
sentences, so it seems to me that still was future time reference. The
conveyed information was exactly the same and using the present tense is
just a grammatical, not a conceptional difference. I'd say nothing in
grammar changes your experience of time.
Apart from that, I would actually prefer to say "Morgen wird's regnen / kalt
sein" in German... The strong-FTR vs. weak-FTR distinction seems way to
rough to me: English and German both have explicit future tense markers, but
also allow their omission in some contexts, whereas Japanese and Korean lack
future tense markers altogether, yet German and Japanese are both put into