A fine line, but I see the nuance. I imagine that a flight attendant would refuse to handle or serve alcohol not owned by the airline. Refusing to serve it to the passenger owner of the alcohol creates a dilemma of "I won't serve it to you, nor return it to you, I know that you own it, but I'm keeping it for now.". I can see an argument arising out of that. Best avoided entirely by insisting on serving only airline alcohol.
The next line of the same regulation reads:
(b) where no flight attendant is on board, has been provided by the operator of the aircraft.
Though not a factor on a commercial flight, this was a problem when I was crew on a corporate Piper Cheyenne, and the people in the back brought booze aboard, and got terribly drunk, while we flew. The Captain and I declared that we served zero booze, we were flying. The drunken passengers asserted that they were the "operator", and as it was their corporate aircraft, that was a little hard to argue back at them. Ultimately, the wise corporation declared that people onboard were "working", and as such may not consume any alcohol, so it was banned entirely from the plane, though as said, but company policy, rather than regulation.