I have the following question about what can be done within the Techum based on the Gemara on this page.
If one is within their Techum do they have the right to carry within the 2000 Amos? Or does the Techum only entitle one to walk around an entire city and 2000 Amos beyond that. As far as carrying is concerned that would depend on the status of the Techum area. So if the Techum area was a Reshus Hayachid one could carry in the entire area. If it was a Reshus Harabim one could only carry up to 4 Amos.
The reason I raise this question because when the Gemara on 45b talks about the Reshus of water taken from the well I understand it to allow one to carry the water within its entire Techum.
On Shabbos it is permitted to carry only within a Reshus ha'Yachid, as you wrote. The Gemara regarding the water taken from the well, and the rain that fell near the city, is referring to Yom Tov, as is clear from the Gemara there. On Yom Tov there is no prohibition against carrying when necessary, but the prohibition of not going beyond one's Techum does apply on Yom Tov.
Be well!