Nice start. You should however reconsider your choice of words 'high power' when utilizing a 2N2222A transitor.
Here's the "H bridge" with only 2 transistors:
performing the same function as a H-bridge
No, it's not. An H-bridge does not require a split supply. This is a half bridge and does good stuff for sure, but an H-bridge is a very specific circuit topology that is different to what you have.
It's certainly dissimilar with an H-drive in that it isn't designed to be PWM modulated, nor any other digital interface to a microcontroller.