$1000 buys you a damn good bike. You don't need racing level components to have an awesome bike, something like Sora is perfectly good stuff and bikes in that range can be had for ~$800 (sometimes less).

It really depends on how much you are going to ride and how much you value speed. Road bikes will put you in a more aggressive position, which definitely translates to more speed (aerodynamics is a huge factor for speed, significantly more so than bicycle weight). If you are planning on shredding a ton of miles and you want to go as fast as possible then a road bike is a great way to go.

Hybrids are well liked by a lot of people because they are more comfortable to ride. What you sacrifice in speed you gain in comfort (if you prefer a more upright position).

A cyclocross bike would be another solid choice perhaps. It's very similar to a road bike though not quite as aggressive but you get the benefit of having drop bars like a road bike with some added CX-specific perks like wider tire clearance and disc brakes if you go that route (cantilever brakes are also common but I prefer discs).

You mentioned 10 miles being too far for a hybrid, but that's absolutely not the case. You can go as far as you want, you just won't get there quite as fast as you would on a bike geared more for speed.

Sorry I didn't answer your question directly, but it's hard to answer because it all depends on what you really plan on doing with this bike. You can get great exercise on any bike! In my opinion a CX bike is a good starting point because they make great commuters, they can go pretty damn fast, and they can handle gravel, city streets and most urban environments with ease. If I could only have one bike in my life it would undoubtedly be my CX bike (just don't tell my other children that please!)