Not sure I understand...what is the difference?
Id prefer to ask "Are you mixing in stereo?" - that is the only relevant question..
We'll there are undefined terms there: mixing what? stereo what?
There are often multiple mixes happening: monitors, PA, sub-group busses, sometimes a recording matrix, etc.
Mixing in stereo? does that mean using a stereo board with a 2-bus out?
People use different terms for things and what might seem clear to you may not be to someone else. Ever ask a sound guy for a 'soundboard feed' and get a puzzled look? It helps to be really specific.
Asking specifically if the same information is coming out of each speaker is about a straightforward as it gets.
..the myth of 'audience ambience' being desirable.
No myth. It is a very desirable aspect which I work hard to balance with the clear, direct sound in my recordings. I agree with you that 'stack-taping' is often used in a pejorative sense around here which is unfortunate and also a misnomer since recording by the board is simply 'stack taping' from farther away. The real issue is optimizing the direct to reverberant ratio. More specifically, it's balancing all the the direct sound elements (the FOH stacks, front fills, sound from the onstage instruments and amps themselves which is the 'real' direct stuff, maybe delay towers if you are really far back in a big venue) and the ambient sound elements (reflections, reverb, general crowd reaction, specific nearby individuals, other room noises, HVAC sounds, wind, trains, planes, automobiles and the guy in the icecream truck). Without enough ambient information, there is little of the live feel, excitement, and feeling of being in the actual space experiencing the event which is a big part of what makes my live recordings so enjoyable for me. To get the best balance I set up in some unusual places sometimes, and yes, more often than not I like to have a clear, direct line and good proximity to the direct sound sources, which often means somewhere other than right next to the sound board.
Location, location, location.