Prius Turning Partly On Without Key In Car

My 2005 Prius is having issues turning on and off properly. Sometimes when I try to turn on the car, my foot on the brake and the key in the car, it does not turn on. The red blinking car light with the key symbol (key detected dash light) will also disappear when I have this issue. I find that physically getting out of the car with the key, and getting back into the car with the key helps reset it and I am usually able to turn the car on this second time. I have gotten into the habit of putting the key fob right up to the start button every time I turn it on because that seems to help the issue.

Yes, I did try replacing the key battery, and that has not solved the issue. I also replaced the Prius 12V battery over a month ago. It was driving fine for the first few weeks after replacing the battery. It's only this last week that I've started having these issues turning the car on.

The car also does not always stay fully turned off. I drove my car this morning, parked at my office, and then went back out an hour later to get something only to find that the dash lights were on and the Start button was red. The engine was not audibly running. I know that I turned the car off when I parked. I even looked in the window after I parked and got out of the car to make sure it was off because I've been having these issues. A couple things to note- I turned the car off while having my foot on the brake, and I'm going to test out not doing that anymore. I also did not lock my car and I am wondering if that has anything to do with it turning back on (although it's obviously not normal for that to happen either way). I also notice a small quirk, that when I put my foot on the brake to turn the car on and press the Start button, the times that the car does not turn on, the brake will pop back up from the pressed down position I had it in.

Any thoughts?