That other question quoted a different part of the rule book.
Support is cut when you are attacked UNLESS that attack is from the territory you are supporting an attack against* which neither question asked about, but is very important.
The exception** to this is, that no matter what, if a unit is dislodged, then the support is cut anyway.
So, imagine 'the support' we're talking about is: Denmark Supports Holland - Kiel.
(as usual we will assume Holland - Kiel)
Now we add: Kiel - Holland. (this doesn't cut the support, Holland is now beating Kiel 2vs1)
Now let us add: Munich - Kiel. (Here we assume Munich is trying to defend Kiel by moving a second unit there. Still we have 2vs1 in Kiel, and now we also have Munich fails to take Kiel, because that original support wasn't cut)
Finally we add: Baltic Sea Supports Kiel - Denmark. (Now Denmark is dislodged, so the ** exception comes into play, the support is cut, Denmark retreats, Munich and Holland are free to bounce in Kiel 1vs1)