What points was Achitofel adamant about?
Malbim: (a) They should pursue immediately, when David is tired, before he rests and prepares himself for war. (b) Do not take all of Yisrael, only 12,000. (c) Avshalom should not pursue, rather, Achitofel. (d) They should not pursue during the day, rather, at night. The coming verses explain why.
Why did he say 12,000 men?
Radak: From each Shevet, 1,000.
Why did he say "David" (without calling him king)?
Radak (3): Due to his hatred of him, he disgraced him, and counseled to kill him.
Even David (15:19) called Avshalom "king" (Radak there - because Yisrael wanted to make him king). We can say that Achitofel holds that now, David is a commoner! (However, in verse 2 he called him king - PF.)