The distinction between objective and subjective collapses under your usage.

No, the distinction is clear. If your assigned probability is something that someone else can reproduce using the same steps given the same information, then it is objective (yet contextual). Your examples each have a clear reasoning behind the assigned probabilities, they're not just opinion-based assertions.

Bob has a very simple algorithm to output probabilities. He just answers 50/50 for any yes or no question. This is reproducible. Is this objective?

This doesn't strike me as a good example of reproducibility in this context. Let me offer another one. Let's go back to your example of the 66% weighted coin. Given the physical properties of the coin, different people could independently come to the same conclusion that the probability of heads is 66%. I would describe this as an "objective" probability, as it's a nice representation of the available information, independently reproducible by different people, given the same information. It's different than "Bob arbitrarily decides that any yes/no question has 50/50 probability", which is inherently subjective.

No two people ever have the same set of information. And very few cases are even as clean as the coin case.

A more typical example is using polls to predict elections. 538's model ended with Biden around 90% to win. Andrew Gelman's model at the Economist ended with Biden around 95%. Do either of those represent objective probabilities?

Or take weather predictions. Per https://www.metaculus.com/questions/4617/will-2020-be-the-wa..., Berkeley Earth gives a 16% chance to something that NOAA gives 29.2% to. Is either of those an objective probability?

I would say no, and I think that just because two people happen to agree on a number in a particular case doesn't make it objective. If you want to use the word objective, I don't have any particular objection. I'm not here to fight over words, and none of these words are really well defined enough to be worth fighting over. I don't think it's useful to think of probabilities as "real" in any sense.

Yeah, that's objective. It's just not a great model.