No. Nanotechnology is not only about mechanical manipulation. In fact, sometimes it's misleading to speak of mechanical manipulation on the nanoscale, as there isn't much of a connection to classical mechanics.
Nanotechnology is pretty much every structure on the nanoscale, basically everything with dimensions that can reasonably be expressed in nanometers.
Because of that it doesn't make much sense to ask for the difference between nanotechnology and chemistry, as chemistry can be nanotechnology if it is done on a scale small enough.
You got that backwards and wrong: "top down" are the fabrication methods where you use litography or other methods to imprint your design on a substrate. Pushing atoms around would fall under "top down". "bottom up" are methods where you let your structure grow by exposing it to different chemicals and use self-assembly processes. Initiating a chemical reaction to get you structure falls under "bottom up". ("top down" also needs some chemistry, but you don't control your structure with chemistry.)