I find that I'm on the more relaxed end of the spectrum. I've noticed from hanging out with a lot of new moms, that I've seen a lot of people who jump everytime their kids fall, whereas I'm more inclined to not make a big deal out of falls and stuff, and just kind of take a wait-and-see approach.
My SO also was surprised to discover her own ability to relax by observing me (this was especially clear one day when I was letting a not-quite-a-walker crawl up onto a chair that was relatively close to the ground and play around). Her first instinct was to go stand ready-to-catch, with her arms out and everything, but when she watched me and then backed off, she and I both discovered this not-quite-a-crawler was actually very adept at getting on and off the chair safely. If SO would have stayed in the ready-to-catch position, her instinct would have been to think the poor kid was falling when she was actually about to get down. We all learned something that day.
I'm not too worried about bruises, scratches, bumps, and even broken arms. I do, of course, draw the line at any exploring that would include life-threatening or more serious risks. I would encourage you to do what you need to do to feel safe about what is going on with your child, while being cautious not to step in when it isn't truly necessary.
I've said that, but I also want to say that it *is* important to keep little ones (especially just after they start walking) generally in sight, particularly if you're in an unfamiliar place where you aren't aware of potential hazards. I don't think that is overprotective.
Also, as others have said, you are learning as much as your kido, and I wouldn't question too much about your style. You'll do what comes naturally, and if you find it too exhausting after a while or if it doesn't work or if you get more comfortable, you'll change what you are doing. No need to stress. You may find yourself more or less relaxed with your next child (provided you are planning to have one). It would depend on what your experiences were the first time around, the personality of your second kid, etc. What you are doing for now sounds fine.
I'm pro-adoption reform, but not anti-adoption.