Thanks for linking to this very thorough treatment of the question. As Cardinal Law explains, Masonic membership is still banned, but no longer leads to automatic excommunication.
Turning back to the question that started this thread, it would seem that any prohibition on membership in a GLO would depend entirely on the specific GLO in question. Certainly, no GLO is likely to engage in the political agitation or anti-clericalism that has characterized Freemasonry in Europe and South America at times in history. And I think that few, if any, GLO's are likely to present the thorough and detailed philosophical system that Freemasonry entails.
That said, it is certainly true that many GLO's, fraternities primarily, were influenced by Freemasonry and turned to masonic ritual for inspiration in crafting their own rituals, as can be seen at
this site. Absent any specific prohibition on a specific GLO, then one must use one's best judgment as to the compatibility between the teachings/principles of the GLO in question and the teachings of the Church. I think investigation will show that this was one of the reasons for the founding of specifically "Catholic GLO's" -- an prospective member would be sure that there would be no such incompatibility.
My 2 cents.