Traffic is generally bad in the city I live in. Years ago I was driving when someone accidentally cut me off. I started to get mad when suddenly the person who cut me off held up a sign I could see through their back window that said "Sorry!". The car belonged to a business and they gave their driver a sign on a handle they could hold up if they needed to. It immediately diffused the situation and made me smile. Little things in life really help.

Warning lights. In France people do a short blink with the hazard lights to say thank you or sorry.

“sorry, I screwed up. Excuse me. I just need to slip in there.”

My problem with this is that the screw up is the slipping in. In other words, accidents happen and there's nothing wrong with apologizing for something you already did by accident -- but in this case, it's preemptively apologizing for doing something on purpose that you know is wrong, cutting someone off because you already were late/past where you need to go.

If you cut someone off because you didn't see them and you thought it was safe, that's a good reason to apologize. Mistakes happen and it's good to take accountability.

If you're seeing that it's not really safe and the other guy is going to have to slam his brakes because you're going to consciously on purpose cut him off, then yeah you're just being a shitty driver on purpose. Apologizing for something you knowingly did on purpose and will do again? That just makes you a liar on top of that.

Various horn sounds is a good idea. I always wanted text messaging displayed on the vehicle.

I used to have a car with a very aggressive horn and never wanted to use it because it did sound like I was cussing something out. I couldn't get the light "beep beep" of communication, only the FUCK OFF ASSHOLE honk.

Now have a different car and can use the horn as a communication by short tap tap.

But the wave is the apology. I have had people accidentally cut me off and when I swerve to pass have gotten a wave and mouthed sorry.

"A bad driver never misses their exit"

Sometimes you make a mistake and miss your exit or end up in the wrong lane. That's on you. Re-route, follow the lane you're in, take the next safe turn/exit/lane change to get back on track; you don't just cut in line, forcing others to correct for your own mistakes. That's a great way to create unsafe conditions, end up in accidents, and hurt people. It's not just about peoples ego/anger, it's genuinely dangerous.

Your father doesn't need a 'sorry' button, he needs to correct his poor driving behaviour and not endanger himself, you, or others on the road.

There's already a gesture 🖐️ and there's blinking your hazard lights.

Or your dad could accept that he's going to travel a tiny bit further than he originally intended, and reroute with a u-turn, or a block circle, or a parking lot turn, or something similar. No invasive lane change. No need to apologise for an invasive lane change. No need for a special horn to communicate an apology for an invasive lane change. You don't need to apologise for something you don't make into someone else's problem.

I flash my hazard lights a few times.

You deserve to be able to retire after someone from Shark Tank takes your idea and makes it mandatory on all vehicles.

In situations where you're basically saying, "thank you for being a better driver than me," opening your window and putting out your hand with all fingers fully spread is the correct gesture. For me at least, it dissipates my anger when someone has cut me off, because at least they're acknowledging and apologizing. I usually assume the reason is because shitty GPS gave them impossible directions.