Yes, it is the Buekenhout definition, when the two points are equal (and no hypothesis with the points distinct). The Moufang set people tend not to cite people who worked on the topic before Tits, probably because they didn't use the terminology of Tits. (So, for instance, John Faulkner's early work is also not often cited.)

The paper has five citations:Hirschfeld's book, the Buekenhout-Cohen book, J.Tits, Twin buildings and groups of Kac-Moody type 1992,PM Johnson Semiquadratic sets...1999 H Van Maldeghem Moufang lines 2007. None of them really take the idea much further. The last one is essntially studying the translation line in the Luneburg plane from this perspective.

(continued) This is for the restricted definition where there's only the hypothesis when the two points are equal. Buekenhout has replaced Desargues (little Desargues in the restricted sense) by the symmetry it induces on a line, and a general line in a non-Desarguesian plane won't have that symmetry.

@MikeShulman A line in a projective plane is a projective line in this sense if the plane is a translation plane with respect to this line. Most lines in non-Desarguesian projective planes are not projective lines in this sense.