By using a slightly different offset you get a slightly different nonlinear transformation, and one that may work even better.

That seems pretty unlikely. There's always some subjectivity to the details of coding and transformations, but what constant you add to make logs behave is not one I have ever seen materially change anyone's analysis; I don't think this bikeshedding makes a lick of difference. Again, if you think it does make a difference, I have provided all the code and data.

For example say instead of denominating everything in dollars you'd denominated in cents (and added 1 cent before logging). Then everyone would move up the graph by pretty much log(100), except the people who gave nothing, who would be pulled further from the main part of the graph. I think this would make your fit worse.

Maybe. But would it change any of the conclusions?

In this case we might think that people tend to give something back to society even when they don't do this explicitly as charity donations, so add on a figure to account for this.

...why? One 'gives back to society' just by buying stuff in free markets and by not going out and axe-murdering people, does that mean we should credit everyone as secretly being generous?

My feeling is that $1 is probably smaller than optimal under either interpretation. This would fit with the intuition that going from donating $1 to $9 is likely a smaller deal at a personal level than going from $199 to $999 (counted the same in the current system).

Disagree here as well. As you already pointed out, a more interesting property is the apparent split between people who give nothing and people who give something; someone who gives $199 is already in the habit and practice of donations just like someone who is giving $999, while going from $1 to $9 might represent a real change in personal propensity. ($1 might be tossing a beggar a dollar bill and that person really is not a giver, while $9 might be an explicit donation through Paypal for a fundraiser.)