This sort of goes into a meta-halakhic question of when the circumstances behind a statement in the Gemara change, whether the halakhah changes, or whether we assume that the rabbis in the Gemara were reacting to their circumstances but setting rules that are independent of them. There are cases where the halakhah definitely changes (like assuming fires on Shabbat are only threats to property and not life), there are cases where the halakhah definitely doesn't change (like medicine on Shabbat, really most gezeirot), and there are some that get debated on occasion (like mayim achronim). There are unsettled issues in this area. But usually the initial assumption is that the Rabbis used their assumptions in setting up davening, and now that they have, the liturgy they instituted is relatively fixed even if what they had in mind has changed.
This is discussed in later halakhic texts. The Shulchan Arukh, OC 110, rules that one does not say Havineinu in the wintertime. As does the Rambam (MT Hilkhot Tefillah 2:4). Some of the commentaries on Shulchan Arukh do not mention any difference davening from siddurim (Mishnah Brurah, Taz). The Kaf HaChaim there notes that if one erred and did havineinu during the rainy season, one still fulfilled their obligation. The Magen Avraham there says something about one who is sure they won't err, which might apply to this situation, but I can't quite parse what he's saying.
edit: However the Koren siddur notes permit saying havineinu in wintertime. But I'm not really sure a siddur would prevent getting mixed up. I don't think most people read the siddur so slowly that they'd get up to ten tal umatar and think about whether it's winter or not. People get confused and say/don't say the correct one with siddurim even without the havineinu confusion.
The Chayei Adam rules similarly (Klal 24, part 30) that one can say Havineinu in a pressing situation, but not in wintertime.