Just a few thoughts of a co-Mason. I don't know how things are in the USA (or the other countries of people reading this), but in my country people are pretty clear about which organisations have 'visiting agreements' and which do not. Members of the Dutch Grand Orient visit my lodge frequently, but not officially. This (of course) brings funny situations with greetings that cannot be returned, but it works well enough. Regular Masons are never invited, but those who which to visit are welcome and as far as I can judge do not make a secret of that in their own lodges. They simply think the division is old-fashioned and that us co-Masons do the same as they do.
Now there is the argument of different rites, but this would be the same if a Mason visited a regular, foreign lodge too. Besides, experiencing another rite enriches ones view on ones own, not?
In Belgium (I have heard) all male organisations besides the Regular Grand Lodge accept visits of the mixed obediences. This is something I will probably go and find out once I am raised Fellow. I wonder how this would work for my girlfriend. I expect that I will be able to visit and she does not, but time will tell.
The bottom line, over here there are 'progressive' regular Masons who enjoy visiting mix lodges and we welcome them. I doubt it will harm the good men
And to answer the initial question: yes and no. Yes who do the same as you. In fact, our rites are often based on yours. The differences there are no bigger between us and you than between different regular lodges.
No, the organisation that first used the term "Freemasonry" (but forgot to copyright it) has terms we do not live up to which makes us another organisation that uses the same name. Not all of my brothers and sisters will agree with me about that, but in my opinion this is just the way it is and I wonder why it actually matters.