This is one of my favourite websites. It cries out "I have my own fashion sense, and you'll just have to accept it while I provide you with a truckload of useful information."
Sengelspiels website has been around forever. I guess he created it like that, back when many websites looked like it and never bothered to changed it.
The information is useful, but rudimentary. The thing to remember about phase in acoustic systems is that, as long as you're driving your transducers (speakers) at the correct polarity (+/-) vs. (-/+), you're probably good. Acoustic frequencies are low so the lengths of acoustic waves are hundreds of meters long. Small phase differences from a local transducer become irrelevant. Phase differences between transducers distributed over a larger area can be a concern, and can be automatically corrected for a specific location with modern DSP/equalizers.
Acoustic wavelengths vary from 17m for bass frequencies to 17mm for the very high end. So phase is absolutely an issue.
If you want to hear this for yourself, play sines wave between 1k and 10k (if you can hear that high) through some stereo speakers. Move your head and you will hear some very clear diffraction peaks and troughs.
You only really lose stereo location below around 200Hz, which is around 150cm.
With domestic speakers, room reflections and other influences make it hard to hear phase issues. But in a studio you want to hear what's happening as clearly as possible. Some brands - like ATC - put quite a bit of effort into keeping phase coherency.
Phase can (and will) become an issue (comb filtering depending on the listening location) when you play phase coherent signals over more than one speaker. For most home applications this might not be a big issue tho if you don't have a totally unconventional speaker setup.
If each speaker plays something else you are good (depending on how uncorrelated the signals are).
If you need to add extra speakers to "extend" an existing set of speakers (e.g. on a huge stage) the phase correlation of those extension speakers will be critical. That means, ideally you want the extension speakers to "fire" exactly at the moment when the wavefront of the original speakers hits. That means you have to delay the signals for the extension speakers by the time it takes the wavefront to travel from the original speaker to the extension speaker (influenced by distance and air temperature).
One thing to consider about phase as well is that the same thing in reverse applies to recording as well.
