Sport Hybrids For Commuting and Exercise?
If you're dealing with rough roads, a bike with slightly wider tires will be more comfortable, less likely to throw you with small obstacles/ruts, and probably have a wider selection of puncture-resistant tires.

Most road bikes are designed for very narrow tires and can't accomodate anything much wider. Narrow tires on rough roads are more likely to get caught on small ruts (etc).

You could always put the widest puncture resistant tires possible on your old road bike (maybe also a tire liner, puncture resistant tubes, etc) and pack a decent repair kit (2 tire levers, 1 tube, a few patches) into a seat bag.

I think the definition of a "sport hybrid" is fairly flexible (vague, inconsistent).  "Hybrid" almost always means a bike with flat handlebars (like a mountain bike), but with larger thin wheels like a typical road bike. A "sport" hybrid is one that's a little more geared for speed than comfort (possibly even the same as a "flat bar road bike"), so maybe has you leaned a bit more forward and has some higher end components and probably no suspension stuff.

For a less than 5 mile commute on relatively flat ground, any bike you like will do, as long as it's capable of carrying whatever you want to carry and ride comfortably over the roads. For your longer ride you're more likely to want the drop handlebars, mostly to have more positions for your hands, but also to go faster. You can add extensions or replace the bars on a flat-bar bike to get more hand positions, though.

For the somewhat longer rides, if you want to be able to go relatively fast and you have to deal with bad roads, I'd look at:


Sport hybrids
Cyclocross bikes (made for racing on mixed trails/mud/pavement)
Touring bikes (road bike designed for long trips in possibly terrible conditions. wide tires, tough frame)
"Commuter" bikes (hybrids optimized for commutes).
any number of more basic less specifically named "flat handlebar road bikes" or "hybrids"


For all of those, you're likely to want to replace the stock tires with something more puncture resistant (for the glass you don't see), and maybe have those be wider tires, too.

From your description of liking drop handlebars, I'm thinking a cyclocross bike might be the best option. Cyclocross frames tend to be very similar to road frames, but can handle wider tires. There's other less important differences that could be asked in another question. Probably won't come with handlebars as low as many modern road bikes, but that is likely to be fixable. The cheaper cyclocross bikes often have the stuff to make it easy to mount a rack and/or fenders, too.