Dang, just realized I misread the title. I thought the question was who would send 100 settlers or more (on a single ship) first.

Reading with fresh eyes I see it is "Who will have sent the plurality of the first 100?"

is a spaceship built by SpaceX with NASA painted on the side a SpaceX spaceship or a NASA one? is it Prob-in-proportion-to-logo-size?

How would this have resolved for "Conditional on humans landing on the moon by 1970, whose spaceship will send more of those to land by 1970?"?

Hmm. There's a lot of criteria that could disagree:
Who paid for it?
Who built the parts?
Who did the final assembly?
Who crews it?
Whose launch facility is used?

I'm going to weight each of those equally.

For the 1970 question this would have resolved to 80% NASA and 20% Other because NASA did 4/5 of those things but subcontracted the parts to a zillion private companies.

If NASA buys a Starship off the shelf and uses their own crew and launches from Boca Chica this would resolve 60% SpaceX 40% NASA

@JonathanRay Boeing was the prime for the Saturn 5 and North American for Apollo, so with the above list assigning the balance to "other" makes sense

@Adam (North American merged with Rockwell, were acquired by Boeing, were spun off as rocketdyne to Pratt and Whitney, who sold them on to Aerojet)

@JonathanRay What counts as "Launch facility"?
If SpaceX launches a rocket from Kennedy Space Center, is that NASA or SpaceX? NASA owns KSC, but SpaceX leases several pads, and those leased pads is where it might launch from. If I lease a car I still regard it as my car while I'm leasing it.

SpaceX or it probably just won't happen