Philosophers talk about deontology and utilitarianism. From Gabriela Guzman:
Deontological ethics is an ethics system that judges whether an action is right or wrong based on a moral code. Consequences of those actions are not taken into consideration. This ethics system is intended to be precise and by the book. Doing the right thing means to follow proper rules of behavior and, by doing so, promoting fairness and equality. . . (emphasis added)
In the other hand, utilitarian ethics state that a course of action should be taken by considering the most positive outcome. This ethics system is more accurate when it comes to addressing complicated situations, which solutions are not as trivial.[This is a very brief pair of definitions. For more nuance, check out the link above, or find other sites that discuss it.]
Roughly, using this way of seeing the debate, one could argue that those calling for impeachment now - because they see the president as having committed high crimes, misdemeanors, and treason - is the right thing to do. It doesn't matter whether the Senate votes for impeachment. Doing the right thing is what is important. The law/constitution was broken, so action must be taken. At the extreme case would be the swimming referee in Anchorage who disqualified a female swimmer because the rules required the butt cheeks to be covered.
And those calling for a careful calculation of how this is going to play out in the Senate before impeaching, could be seen as utilitarians. What's the point, they'd say, of the House voting to impeach, if the Senate does[n't] vote to convict?
But, of course, life doesn't settle into neatly articulated categories. One could argue that demanding impeachment hearings start now, is the best strategy to get rid of the president - either via impeachment of the 2020 election. Impeachment hearings give the House the power to investigate the president's actions, to get documents, tapes, and to compel witnesses to testify. That process itself, they would argue, could lead to revelations that would swing enough Republican Senators to obtain a successful conviction in the Senate. And, even if that doesn't happen, it could reveal enough to help Democrats take the presidency in 2020. Which would put those folks into the utilitarian camp.
Rep. Pelosi, who clearly represents the chief utilitarian in the original scenario, would argue that getting rid of Trump and restoring the US to a nation of law, is the ultimate goal. If we go the impeachment route, we need to win, not make a show of ideologically pure failure.
As I think about this, I'd say Rep. Pelosi fits fairly neatly into the utilitarian box. But I suspect the impeach now faction is made up of folks who are clearly deontologists and also utilitarians, who see impeaching now as the path to the best overall outcome. And some may feel that impeaching now is both the right thing to do and the most likely path to accomplish their goals.
These splits among people who ostensibly hold the same political beliefs (or religious beliefs) is not uncommon. Humans probably line up somewhere on a continuum from Deontology to Utilitarianism. Those on the ends of the scales aren't likely to budge.