Agree with Kdelong's advice to hit a well-stocked cycling shop--preferably one with 3-4 different makes of bike at least, and of course a good selection of different types of bikes.
I was in a similar situation to yours almost exactly a year ago-- I'm coming up on 54, and after a life time of running/jogging and an unfortunate accident when I was much younger, things finally caught up with me through my back. My running days were over, and it was getting to the point where I was having serious trouble walking more than a block or two.
I went bike shopping for the first time in a long time--last bike I owned was some kind of Schwinn mountain bike derivative that I toyed with briefly in the early '90s...before that, I think my last bike was when I was around 12, LOL.
Needless to say, things have changed a lot--and it's all for the better!! There are all sorts of variations of bikes today, each with their own primary emphasis (road, trail, fitness, commuting, etc.) but most able to do more than one thing well.
I initially went with a hybrid for much the same reasons you mention--concern over riding posit on a road bike, and uncertainty whether my back would permit it. After about a year though, I decided I wanted to try a road bike, as we just found out we're going to be moving to a very bike-friendly location. I was originally gonna trade my hybrid in, but decided to keep it for general running around, less-than-ideal surfaces, etc.. Good news is the docs have done some good work with my lower back issues, and I'm not only able to walk distances again, but riding is a real pleasure.
I think you can achieve reasonable fitness goals with a bike--it's the primary reason I ride, but as suggested above, you need to be able to devote a bit more time to it. I'm currently mixing cycling with time on the gerbil machines in the gym, but plan on doing mostly cycling starting this summer when we move.
You can buy yourself a very well-made, nicely equipped hybrid or road bike for the budget you have in mind--and likely have enough left over for a good helmet, bike shorts, etc..
Happy hunting!!
Bill