American Cancer Society is a good example of an organization that is clearly, unambiguously political, but which most people probably don't want banned.

Ultimately, every position you can possibly advocate - including the position that you want people to purchase something or donate to someone - has packaged with it some vision of how society might best operate. In other words: in a society where decisions are made in a political way (which is a good thing), every message is political. Every message.

Sounds like the American Cancer Society can advocate for itself but not for candidates or policies.

Fine:

Donate to fight cancer

Not fine:

Donate to Candidate X to support cancer research

How about ACLU or Planned Parenthood?

same apply. ACLU can ask for donations or talk about their services, like representing people who can't afford representation.

PP can as well. They can't say "vote pro-choice" but can advertise that they provide women's health services including abortions.

Basically: you can talk about what you do that isn't related to elections or votes in congress.

Can the NRA ask for donations?

My guess is that it depends how it is phrased, the imagery and the landing page. "Donate now", ok. "Donate now to keep NRA working for you", ok. "Donate now so we can fight policy XYZ", probably not.

This is really interesting, because it seems that in some cases (ex. NRA or EFF) it doesn't actually matter what the ad copy is. It's very clear why they're soliciting donations (to lobby for their ideological viewpoint).

So in your example, I think it's absolutely clear to most adult Americans what "Donate now" means. It means "Donate now so that we can fight for your gun rights" or "Donate now so that we can fight for a free and open internet".

I do worry that Twitter is just ratcheting up the difficulty they're already facing with political speech on the platform.

Those are a civil rights group and a health care provider. If your politics are opposed to civil rights or health care then you may consider them "political" but they aren't. Not by any reasonable definition.

I'm pro-choice, and support/donate to Planned Parenthood, but you have to be willfully obtuse to not accept that abortion is a political issue — 40% of the country thinks that it should be illegal.

I also donate to ACLU, but you have to be willfully obtuse to not accept that it predominately backs left-leaning policy viewpoints, and vocally so — again, fundamentally political in nature.

If an organization lobbies to receive taxpayer funding, or lobbies to change laws and legal rulings, they are political. Whatever else they happen to offer doesn't cover that up.

Both organizations have PACs. Their PACs should be covered by any ban on political advertising.

What good or service is "Donate to fight cancer" advertising?

Sounds like a Kickstarter for a cancer cure, it's a legit service.

I would think that yes, it means they couldn't advertise.