I'm a pretty non-musical person, and time signatures have always confused me. Unfortunately, this explanation left me just as confused. I can hear the difference between the two sample beats but I cannot figure out how to generalize this to other time signatures beyond the 3/4 and 6/8 described here.
3/4 is ONE two three, with a foot stomp on the ONE. It's a dance beat - STOMP skip skip.
4/4 is so common it appears in a lot of variations, from the bass drum beat of EDM to all the different rock and pop variations. The usual - not infallible, but strongly suggestive - tell is how often the chords change. Failing which, there's often a unique rhythm marker at the starter of each bar.
6/8 is skiddledeediddledee - often used in folk and/or ethnic music. It's a much lighter, more skittery feel.
You can generalise the /8 time sigs to 9/8 and 12/8, although they're more rare. They're often used to signify something exotic, alien, ethnic, or maybe a bit spiritual.
Like this, which you've probably heard, and is in 12/8 when it eventually gets going. The clue is in the phrase length:
https://www.youtube.com/watch?v=jLnL63cXmD8
Exotic sigs like 5/4 and 7/4 are much less standardised, so they're open to interpretation. You can play 5/4 as 2+3 or 3+2 or 4+1. Or just as 5.
Likewise 7/4 (listen to the snare):