Drivers DO get notified of your tip, and of course, we see it in the ride detail. Sometimes the notification comes several hours later, but we have no way of knowing if that's just a glitch or if the rider tipped after the fact.
Don't worry about the driver not believing you about tipping in the app. It's pretty obvious which riders are sincere and which ones aren't.
Also, just FYI neither company takes any percentage of in-app tips. The driver receives the full amount...except...
With Lyft, they count in-app tips as revenue to the driver for the purpose of guarantees/bonuses. If a driver is working on a guarantee of a certain amount, an in-app tip actually reduces the amount Lyft will have to pay the driver. The driver gets the money, but it actually works against them because it offsets bonuses Lyft would pay otherwise.
A lot of drivers freak out when they learn that, but it's right there in the Lyft guarantee/bonus promotion info. Drivers just don't read the fine print, and that's a fundamental mistake with anything Lyft. Gotta read the fine print. Details matter.