Ten minutes late with an honest “I left late” beats any traffic story.

The plain version of lateness is almost always received better than the elaborate one. Everybody has been the person waiting, and everybody knows which excuses are structural. When you name the real cause, there is nothing for the other person to quietly disbelieve, so the whole thing takes ten seconds instead of becoming a small negotiation.

Next time, send the message before you arrive and give the true reason without decoration. Then change one thing about tomorrow so the message is not needed. Honesty about the cause is also the only route to fixing it.