This may sound like a distinction without a difference, but it is explicitly illegal to give money directly to politicians in the United States. Lobbying is giving money to their campaign committees, which get them elected or re-elected. It is also explicitly very illegal for campaign committees to use their funds to do anything for the candidate -- there are a huge[0] number[1] of politicians[2] who have been severely[3] penalized for flouting campaign finance laws. Further, there are severe restrictions on what the campaign committee can use their money for, which is largely restricted to travel for the campaign, advertising, and paying campaign staff.
These are just some recent examples of federal politicians who have run afoul of these laws, but they apply at all levels of government, as well as to staff (both campaign staff and the official staff of elected officials).
Yes it is totally a meaningless distinction to me. On the level of “this is not gambling, because we use a virtual currency (that can be purchased with dollars)” that is prevalent on mobile video games these days. It is gambling, just with extra steps. And it is bribery, just with extra steps.
These are just some recent examples of federal politicians who have run afoul of these laws, but they apply at all levels of government, as well as to staff (both campaign staff and the official staff of elected officials).
[0]: https://en.wikipedia.org/wiki/Duke_Cunningham
[1]: https://en.wikipedia.org/wiki/Duncan_D._Hunter
[2]: https://en.wikipedia.org/wiki/Jesse_Jackson_Jr.
[3]: https://en.wikipedia.org/wiki/Michael_Grimm_(politician)