Military conquest results in the highest score.

It doesn't matter who declares the war from a score perspective but it does change relationships which might make it more difficult to win. It's really not so much who starts the war as to who ends it. If you take an opponents last planet then you are performing genocide regardless of who started the war in the first place. Sometimes with sufficient military might you can get an opponent down to their last planet and then offer them peace but include the demand for their last planet as part of the peace deal. For some surprising reason the AI will sometimes accept this which eliminates that AI but doesn't count as genocide.

When you perform genocide you get tagged with a negative relationship factor "tendency to conquer others" this makes it more likey that other AI's might declare war on you before you are ready. I've also seen cases where the remaining AI will form an alliance because of this and so if you attack one then you will be at war with all.

This may or may not be an issue depending on how strong you are at the time, however this is easy enough to avoid. Simply get the AI down to his last planet at which point they will almost always accept peace and then move on to the next victim. Once you get to the point of not caring whether your remaining opponents ally against you then you just finish them off at your leisure.

There are probably some negative effects of being the one to declare the war. This doesn't affect score but it may contribute to being tagged with the "militaristic" negative relationship tag and it also may affect the AI's willingness to accept a later peace. If they start the war they are more willing to stop if it goes badly for them than if you started it.

However it's easy enough to get them to declare war on you instead of attacking them. Simply park a few strong fleets of ships accompanied with a fleet or two of transports next to an opponents planet and you usually won't have to wait more than a few turns until they declare war on you.

I usually don't bother worrying about who declares the war but I do worry about performing genocide. I do like I said and eliminate an AI down to their last planet and then give them peace. Once I'm down to a single credible opponent then I usually take out all the remaining single planet AI's at the same time.

As another side note, declaring war also raises the chances of triggering the "secret meeting between others...the war has expanded" event.

If you aren't familiar with it, what happens is that you declare war on Species X, and then Species Y and Z declare war on you.

Getting another race to declare war on YOU seems to prevent this from happening.

I think it also has something to do with how powerful your military is VS. everyone else.

Conquering minors will flag you with the genocide flag as well.