Think you either did the handwraps wrong, or simply hit too hard for them to protect properly.

About the gloves, we use MMA ones where I box for when we go on the bags. But I personally recommend using orthodox boxing gloves, just buy the 25$ pair and make sure they're a little heavy. You won't be wearing them for very long at a time, so it will improve your overall punching speed and power wearing orthodox boxing gloves because of the added weight.

Up to you though, if you feel more comfortable in the MMA ones, buy those, it's all about you being comfortable after all, not what would work the best.