"Anonymous. What exactly does swing catch drift mean, if it is not just another way of counting S Q Q."
It is not. It is one of many ideas people have put forward to try to match the nature of foxtrot action.
"Do you really mean to say you would teach a class not to be aware of the beats. That there is not two beats for a slow and one for a quick."
Well since there aren't to most people's understanding of what actually happens, I would probably avoid getting caught up in the useless detail of trying to explain where the steps fall relative to the beats for slows and quicks. If you want a numerical standard for foxtrot execution, hit your first quick on beat three and just feel the rest flowing with the music - pretty much anyone who is off time will get that wrong, so getting that right will get you most of the way in terms of the strict part. Whears trying to count through the rest of the slows and quicks as individual steps placed to beats will only kill you movement and make the dance very halting and jerky.
In trught, the number one reason why dancers go off time in foxtrot has nothing to do with counting, and everything to do with the fact that they don't have the strength and knowledge required to dance slowly enough to match the music. Once they build that capability, matching the music isn't hard - before the build it, it's impossible.
"Why do you think on a DVD they call out the timing on everything they are teaching."
Because they are TEACHING. To do that, you break things down and structuralize them. But when it comes time to dance, you DANCE with your mind eventully unclutered by all that artificial, mechanical approximation. Perhaps you aren't to the point yet where it is safe to trust yourself to do that. If that's the case, then PLEASE, KEEP COUNTING.
As for why to present the slows and quicks, because it's important to understanding the ultimately amounts of time required for a figure, even though they are not actually danced quite the way that this implies. It's important to know if you have SQQ or SQQQQQ or whatever to understand how everything will fit together, but you don't want to get caught up in the trap of worrying about where the boundary between the last quick and the first slow is - that will only get your into trouble and very awkward movement.