I think you have answered your own question. You have done good research, and it shows that "cigarette butts" is more commonly used than the other alternative.
A "butt" is "the part of a finished cigarette that has not been smoked" (Cambridge). Another term for this, but much less common, is "stub".
A "stub" is "the short part of something that is left after the main part has been used, especially a cigarette after it has been smoked ..." (Cambridge). See this Ngram for their usage. The difference in usage between the two is smaller in BrE.
A cigarette butt may also be called a "snipe" in AmE (came across this from user105719's comment). This is a slang, so be careful if you decide to use it. See Collins Snipe definition no. 3 in AmE.
You can, however, write "leftover cigarette butts" depending on the surrounding text. But it is not something you will come across everyday. See this, this, and this.
What to call the part of a cigarette that does not contain tobacco, the yellow one which may contain some wool or cotton, do we call it "the filter" or "the butt"?
The yellow part is called "butt". The "butt" has a filter inside it and some ventilation holes on it. The outer layer of the "butt" is called "tipping paper". "Filter" is the name of the material inside the "butt".
In addition, what if you have a cigarette & you smoked by 20% of it, then stopped smoking, you will then come back to smoke the rest. What to call a cigarette that has been smoked by 20%? a butt or a leftover cigarette?
If much of the tobacco is left (the white part), then it is not a butt. I would then call this a leftover cigarette, half a cigarette, or something of that sort. I am not aware of a specific term for this.
Both Kate Bunting and user105719 mention in the comments that cigarettes not used at all from a previous party/event would be called "leftover cigarettes". You may also call a cigarette with enough tobacco in it a "partially smoked cigarette".