Well, there is also the possibility was that they were exaggerating the degree to which they actually believed he was a puppet.

Democrats impeached a President over something they didn't really believe was true.

They didn't impeach him over "suspicion of being a Russian puppet". They impeached him because he attempted to abuse his power to coerce a foreign government to investigate his political rivals. Note, even the withholding of the funds was illegal because Congress had approved it. This is without the shakedown tactics that he employed on top of it. Let's not rewrite history so quickly.

Democrats paid foreign spies to generate fake evidence, then used that fake evidence to start an investigation into political rivals during an election year which included but was not limited to wiretapping top level campaign officials. The Ukraine investigation was in direct response to the Russia investigation, they are tightly coupled. They clearly do not have a legitimate problem with his behavior, since they had done the same thing themselves. Additionally, it's not clear they believed the claims in the articles of impeachment any more than the claims made during the Russia investigation. This is not an endorsement of Trump or anything he has ever done.

Well, your idea was that they were acting erratically and irrationally, my proposition is that they may just be putting on a show. Neither alternative is particularly flattering.