2003 very strange starter problem cant figure it out
 

hey guys me and my buddy just replaced his bad starter on his 2003 accord, we started it and it worked great, then i said turn it off and do it with the aftermarket remote start, so he did and it started, we started smelling like a burning smell, and turned it off, and the starter was still spinning while disengaged. we had to disconnect battery to stop it. so i reconnected the battery and as soon as the battery connects ithe starter starts spinning, but with out being engaged, is this the remote start system, or a bad relay or something, or even bad ignition switch. and also if it is one of the above how is it possible for the starter to not engage but still spin

Try unplugging the small wire on the starter, then reconnect the battery. What happens?