I'm looking for ways to communicate to my users that our project is committed to making the world a better place.
Are you sure it's not committed to making some piece of software? I mean, sure, that software can be used to make the world a better place, but is that what you mean? Or - do you want to talk about what the organization/contributors do other than writing software?
That we don't collect or sell the user's privacy.
Regardless of badges - have a clearly-visible brief statement to that effect on your website / program GUI / social media pages / etc. And - a longer statement going into details on your website.
That we are committed to promoting the ideals of Open Source Software and the Open Source Community.
I don't believe there are set ideals. I'm guessing you have a notion of what these are based on the views or rhetoric of some prominent activists or developers.
Are there any organizations or Pledges or checklists, that are widely used to communicate good will to our users?
Bottom line: This is essentially impossible, since even on the matter of what constitutes good will there are fundamental disagreements.
In the business world there are things like the Better Business Beau or B Corp or 1% for the planet.
Yeah, those are totally bogus. I'll take my Evil Corp., jack up the prices by 1.01% , then donate 1% "for the planet". Look at me, I'm now an ethical and responsible business... you know what I mean?
Forget the badges. Show your users what you actually do to better society. Be transparent about your funding, use of funds, clients/organizational users.
Edit: ... and consider licensing your software under GPL or a similar stricter FOSS license, rather than the permissive ones. That's already somewhat of a sacrifice in terms of propagation of your software in favor of buttressing FOSS-only environments.