TIFU by pushing a car, crashing and almost k*lling myself

My mom's car has a gearbox issue whereby it does not reverse properly. This has led to a few awkward situations like pushing it uphill out of driveways and my mom driving forwards over concrete dividers because she couldn't reverse out of a car spot. However, while these situations are slightly annoying, I have never learned my lesson in a way more dangerous to property and lives, or as expensive as my fuckup today.

For context, I (22M) live with my parents and our house is at the top of a court. My girlfriend was heading home for the night, but she was parked in my mom's spot at the top of the steep portion of the driveway, and my mom had blocked her in, parking at the relatively flat section at the end of the driveway. No biggie - let's go swap the cars. I try to reverse mom's car and it just revs, doesn't move. Usually she needs to park on the incline and let gravity get it going, but she had parked with the back wheels firmly in the gutter.

Being young, naive and inexperienced, I have only ever had to push a broken down car once and the person I was with opened the window and steered from the outside. The crucial difference, and the heart of this fuckup, is that that was pushing forwards on a flat section of road, not reversing downhill. Nonetheless, recalling this experience, I said "Let's give it a push," to which my girlfriend responded "Do we need to go get help?" Severely underestimating the laws of physics and overestimating my ability to steer a car backwards through the window downhill, I responded, "No, I'll just turn it and it should stop moving when it goes sideways since it won't be facing downhill."

So I stick it in neutral, open the window, and get out. We give it a few pushes to build momentum and get the back wheels out of the gutter. We get it moving pretty easily and now is my time to steer. While I AM able to turn the wheel, I am NOT able to lock it out fully sideways, and the car continues down the hill on just a slight angle, picking up more and more speed. It is now heading towards a neighbour's house.

I try to correct the steering back towards the middle of the road (What was the plan here, asshole? Steer it all the way down the hill into a nature reserve? A house?) But I over-correct, straight into my neighbor's parked car on the other side. There is a tremendous BANG and pieces of metal and plastic go flying. The neighbor, bless him, comes out in his pajamas and is more concerned that I am OK. He tells me to get some sleep and we'll figure out insurance in the morning.

Looking back, I don't know WHAT I was thinking not having anyone in the car controlling it to hit the brakes, or at least pushing it from inside the door where I could hit the E-brake. Luckily no one was hurt, I didn't run myself over and the car didn't end up going through someone's house. Insurance will take care of it, and I will be taking full responsibility.

TLDR: pushed my mom's car downhill into a neighbor's car.