There are various t'fillot/b'rakhot/piyutim that make controversial or debatable claims or requests; for example:
- Birkat hamazon includes the line na'ar hayiti, gam zakanti, v'lo ra'iti tsadik ne'ezav, v'zar'o m'vakesh lakhem ("[…] I have never seen a righteous man abandoned, his offspring begging for bread"); some communities perceive this as blaming the needy for their situation, and simply remove it.
- For that matter, the first part is obviously untrue when recited by a woman, or by a young person.
- The prayer for the State of Israel describes it as reishit ts'mikhat g'ulateinu ("the first sprouting of our redemption"); some communities avoid this claim by inserting shet'hei ("that it may be"), and I understand that some communities choose not to pray for the State of Israel at all.
- V'khol ma'aminim makes a bunch of claims about what "all believe", which I think are unlikely to all be true unless the "all" we're referring to is unexpectedly narrow.
So, when we're praying with a community whose customs don't perfectly match our beliefs:
- Should we skip things that we don't agree with, or aren't sure that we agree with?
- If the answer is that we should skip things we're not sure about, then as a practical matter, what do we do in the very common (if non-ideal) case, among diaspora Jews, that we don't even understand a given passage?
- If the answer is that we should recite even things that we disagree with, then what can/should our kavana be while doing so?
(Note: if the answer turns out to depend on multiple different factors, then I obviously don't expect an exhaustive listing; but some examples with explanations would be helpful.)