First of all, your eligibility is determined on the first day of each month. So if you changed insurance in the middle of April but qualified on April 1, your limit for 2021 would be 4/12ths, not 3/12ths.
The testing period for the last month rule addresses your 2020 contributions, not 2021. The last month rule says that if you are eligible on December 1 (the last month), you could contribute the full year amount instead of a partial amount, as long as you remain eligible for all of the next year. If you lose eligibility during this testing period, your prior year contributions made under the last month rule become retroactively disallowed, and become taxable.
For your current situation, the last month rule will apply to contributions made during 2020. If you relied on the last month rule to contribute the full amount for 2020, then you have a taxable event in 2021 because of the change in coverage. If you were fully eligible for 2020, then you did not rely on the last month rule and you don't have any problems.
The application of the last month rule for 2020 is separate from the excess contribution rule for 2021. You need to remove your excess 2021 contributions, plus their earnings, to avoid a penalty. (You already understand this.) Separately, if you relied on the last month rule in 2020, some of your 2020 contributions are retroactively disallowed. They can't be removed to avoid tax and penalty, you have to pay that. But if you did not use the last month rule, you can ignore this part.
