What exact version of AE? (13.7.x?)
Can we see a screen shot of your preview panel's options?
Does it happen in other projects or just this one?
Have you tried importing that project into a brand new one?
Just tried importing into a new project, same problem, no matter where I put the playhead, the preview will jump to last rendered point and then struggle on.
What it's doing is like the old RAM preview settings.
If you turn off cache before playback, it should do what you want.
Alternatively, change the settings for what happens when you press stop. However, I would suggest making your settings so that pressing 0 on your number pad starts playing cached frames and leave the spacebar stop settings as they are.