Does the source of funding matter in determining academic promotion?
The point of assessing funding is not to ask, "Is this person funded today?"  Rather, it's to ask, "Will this person be funded in five, ten, and fifteen years?" 

And the point of asking about funding is two-fold. Over that time period, will this person be performing high quality work that makes our department and university look better? And will this person be bringing in money that can be shared among other workers through indirect costs?

If you think about that, the various sources of funding do have different levels of attractiveness to the university/department. Someone who is independently wealthy and pays $500,000 per year to support their lab, is less attractive than someone who has an NIH grant that pays $500,000 to their lab, because the latter also pays overhead to the department and helps keep the lights on (literally).  

Moreover, the independently wealthy person may lose all their money, or may simply unilaterally decide not to use it that way after three years. Same applies to a rich philanthropist (or the philanthropist may turn out to be a terrible person that the department doesn't want to be associated with). 

On the other hand, a researcher who has demonstrated that they can get money from the NIH, or the NSF or another long-established government agency, this year, probably has a decent chance of getting more money from them in five and ten years. 

So a source of funding that comes from a stable organization, with predictable and open funding plans, that covers overhead, would be more attractive than sources that are less stable, less open, and less predictable, and that cover less overhead. 

Such stable organizations tend to be government-run (at least in North America, and I think in most of the Western world), but there are certainly private foundations that are just about as good, as far as promotion is concerned.

I don't know of an objective source that could rank the attractiveness of funding, though.