Please.

You should know by now that politicians see events through their own slightly off color and out of focus glasses.

Also, you will be considered more aggressive if you totally wipe out your enemy (as you did when you took their one and only planet).

I always try to leave them several planets when I end the war with them. It makes me appear less aggressive, even if those remaining planets are eventually flipped by my influence. (I try to make sure they have no planets left with overlapping influence)

In the case of a single planet major civ it might be better to simply destroy any ships and starbases, take resources etc. for a time. Then you can offer them a peace treaty. If you do it long enough you will gain more in the end by keeping them subdued while you are researching and building up your forces. Plus you look better to the other civs by not committing genocide.

I don't think approval rating for a planet has anything to do with how many fight when being invaded. That number is strictly a percentage of the population.

And I'm sure you can do the math. If you send 1B troops and half are lost, your population on the newly taken world will be .5B. The only way to make that number larger would be to use Information Warfare to get some defectors before the actual fighting starts (I don't know about Spore Ships, I only have DL). So while you can make up a story about how many survive, they will never be counted as part of your population. They were, are and always will be invisible.

As to why the AI won't talk peace when they are down to a single planet, well, maybe they consider it pointless and would rather go out fighting.

I just played a new game with the same opponents.  This time I left my opponent a planet and the other races were less hostile towards mine.   The only exception was near the end when a race I had created based on the Altarians was in the process of conquering a cluster of minor races.  If they had taken those planets it would have carved in twain my current empire.  So I waited till the minor race's defenders were gone and sent troop transports to take them.  (There was no law requiring me do declare war before hand).

Taking those minor races planets did give me the "Tendency to conquer others label" in the report.  Next time, I may give them a planet that will flip back before invading them.  I noticed that invading a minor race that has treaties with me will not affect my reputation the way that it does if a major race has a treaty with me.  ( I don't have a treaty with them, I just exchanged some tech for their treaties.  Near the end every major race had either an economic or research treaty with me (or both in some cases).

I tried one tip someone suggested in another discussion but that didn't work.  The tip consisted of placing a scout near an alien planet (Korx) and then converting it to a tiny hull with a weapon and no engine and then giving it to another race (Torian).  The idea being that the Torian wouldn't be able to move the ship out of the sphere of influence of the Korx and the Korx would eventually declare war on the Torian.    The Torian was somehow able to move that ship in orbit around his planet despite the fact that it didn't have an engine.   Other races I tried this trick on did the same.  I had no sensors in that area so I couldn't see the ship move (or the "tow truck" arrive).

In this game when someone paid an alien race to attack me, the first time I bribed the Drengin and Korath to attack the Drath as they had bad relations with them anyway.  The second time, someone paid the Carinoids to declare war on me.  So I invaded the Drath homeworld without declaring war and left them their secondary class 4 planet that eventually flipped. 

In the same game, the Korx tried to bully me into giving them credits for their "peace keeping efforts" (they weren't at war with anyone).  The first time I paid it, but they came back for more within a few turns.  The second time I noticed they only had 1 warship so I traded for that ship, then I asked a tribute for the same amount.  (Which they refused to pay).

This will not work as minor races are immune to cultural conquest.

The gifting of a warship w/o engines to initiate wars between opponents was an idea I suggested.  It appears that a better approach may be to gift a troop transport (even with very low troops).  The downside is that the gift receiver may need to have Invasion tech.  That is, I am unsure if one can invade with a troop transport w/o having the tech.  W/o the tech, the troop transport gift might not be seen by the other enemy as a threat.

I suppose one test would be to play a game, avoid getting the invasion tech, and then try to trade for an AI troop transport.  Once in possession of said transport, attempt an invasion.  That does not seem very easy to set up, however.

I like the trick of giving an about to be conquered foe a low pop planet deep in one's own Influence zone just before invading their last planet.  I have used the work-around of carefully choosing which single planet to leave them before peace.

Many times, the AI empire will surrender (not always to me!) when down to their last few planets.  I suppose I could turn-off surrenders, but I like the unpredictability.  Also, turning off surrenders guts the Temple of Empathy of any value, and I doubt the AIs are smart enough to recognize that.