"on" is used for the net in the same way as for the TV or radio. The net is not seen so much as a printed medium which you can physically open and look "inside" - like a book or a newspaper - but as a broadcasting medium. The name "Facebook" is irrelevant. Particular sites are seen as being parallel to TV or radio channels.
I saw the news on the television / on the "Today" programme / on BBC1
I heard the news on the radio / on the "Today" programme / on Radio 4
I saw it on the net /on "Youtube"
I heard about it on the net /on "Facebook".
No - because there it's seen as a "place". A chat room is seen as exactly that - a room that you can go into and come out of when you want. It's metaphorical of course, but what you do parallels entering and leaving a room exactly. So you go "into" and "out of" a chat room. But you see something "on" a site.
Interestingly, forums like this one are sort of half way between the two - so you sometimes see people talking about posting "on a forum" and sometimes "in a forum".
I've written quite a lot of books, but none on grammar... yet The one you mention is very old and I wouldn't recommend it now. It's pre-internet and talks about things like writing telexes...