Lobbyists may engage in PR activity, but that does not make PR activity lobbying.
Lobbying has a pretty clear definition which you are trying to stretch here.
Let me put it this way: Airing an ad about a government rep or piece of legislation is PR or advertising. Using the threat or promise of airing that ad to try to talk a politician into doing something is lobbying.
Without any funding to the politician involved, a lobbyist’s job is to convince the politician of voting a particular way.
A sleazy lobbyist may try money in indirect ways for/against the politician, but a regular one will try to raise awareness through advertising and public pressure. It’s all lobbying.