Can we consider that part in Intolerance: Love's Struggle Throughout The Ages where the general kisses the king (just a fast kiss), as the first gay kissing or not? Because it is not shown in a gay manner, just like a normal habit for those people of that place and time. Or can we, if we consider the actual ideas and concepts about homosexuality?
I haven't seen the film, but given the manner of the kiss you've described I'd say no, it's not the first gay kiss. I'd back this up by pointing to the fact that no gay/lesbian cinema websites, or record collecting books like the Guinness Book of Records, mention it as being the first gay kiss.
I think if you were to take a kiss like that as being homosexual, then any movie where two men kissed each other on the cheek would be gay - an obviously ridiculous concept when you consider how many countries use that as a common greeting.
Instead, I'd point towards Manslaughter, a 1922 silent film directed by Cecil B. DeMille. It featured a romantic kiss between two women in a prison scene (and is listed as being the first gay kiss by The Guinness Book of Movie Facts and Feats) - although I can't cite a reference for this, as it's not available online.
As for the first male gay kiss, that's a little harder. Wings, a 1927 silent film (and only majority silent film to win an Oscar for Best Picture until The Artist in 2012) had a homo-erotic relationship between two men who were supposed to be vying for the affections of a woman (but were more interested in each other). You can see one of their romantic scenes here. However, as the scene shows, their relationship is largely based on the idea of their attraction to each other, rather than by explicitly showing it. In other words, their kiss is a simple kiss on the cheek, much like the one you describe in your original question. The difference however is that they very much meant it romantically, in contrast to the film you cited.
An even earlier film showing a similar relationship was Different from the Others, a 1919 German film, which IMDB describes as: "Two male musicians fall in love, but blackmail and scandal makes the affair take a tragic turn.".
However, like with Wings, no on-screen lip kiss between the males is shown.
Unfortunately, as Wings is commonly cited as the first male gay kiss, I'm yet to find an answer regarding the first lip male kiss, although I'll update this answer if/when I do. But to answer your question:
No, Intolerance does not show the first gay kissing - the first gay kiss in cinema appears to be in the film Manslaughter, between two females.