I personally prefer to ride in hiking boots. It is how I started, and by the time I got myself a pair of cowboy boots, it seemed like I couldn't feel the horse as well. Keep in mind that I am short, about 5 ft, and so boots fit high on me.
I found a good compromise though. They are an Ariat boot, low rise so they are lower than a cowboy boot, they come up to about mid calf. They have a rounded toe, reinforced stirrup area on the top, and are very light. Leather with a rubber sole. They are slightly loose at the ankle so will come off if caught in the stirrup.
If you are going to just go between the cowboys or the hikers though, this is what I would do.
Go for the cowboy boots. Make sure they are well made, and fit properly. Then peruse the second hand shops for a pair of sturdy, well made hikers/work boot. You can get a decent pair for less than $10.
If it turns out you are uncomfortable riding in the cowboy boots, make sure they are fabulous enough that you will wear them on a night out on the town.