score:3
I have done exactly this two years ago and had no problems whatsoever about it. This does not mean you won't have problems but I highly doubt it would be an issue. That is confirming what your airline says.
I.e. I have two first names but routinely only use the first of those. My ESTA only has the first first name while the flight ticket was issued with my full name (i.e. two first names) as it is also in my passport. Neither airline nor immigration nor anyone else even bothered about it.
Given the wording of the ESTA application rules that you cite, I even assume this is the intended behavior.
As @phoog pointed out in a comment below, in the US second and subsequent given names are generally called middle names, so leaving them out of an entry called "first name" would be quite normal.