Heel-and-toe sometimes means different things to different people, because it actually involves several techniques. The key here is that we are often doing two things at once: braking and downshifting.
The first goal in downshifting is rev-matching the clutch plates on downshift vs abusing the clutch. Rev-matching the clutch is what you have to do, period.
The second goal is rev-matching the gears vs "that's what the synchros are for". Here opinions vary. Ross Bentley says it may depend on the car and the track whether it's worth doing.
Double-declutching is the declutch-neutral-clutch-blip-declutch-gear-clutch sequence that does both of the above (if done properly). Instead, you can also shift "straight" (declutch-shift-blip-clutch) to match the clutch revs and let the synchros do their job with the gears.
Now, all of the above is pure shifting action. But because you also have to brake at the same time and your left foot is busy with the clutch, you have to operate the brakes and the throttle with one foot. Which is where heel-and-toe comes in, though depending on the pedal setup it may be more like "side-and-side". (And in some street cars you'll twist your ankle before doing that without modifying the pedals first.)
And then when you downshift, no matter how you do it you can go through gears in sequence or skip. Which is best depends on the situation.
So the gist is, all of the above are separate things you can mix and match. You can heel-and-toe with either straight shifting or double-declutching. Or you can double-declutch without using heel-and-toe if you are not braking at the same time. That's probably not common in racing situations, but on a freeway you may want to go a gear or two down to pass someone. And in any case, you can skip gears (though I'd guess double-declutching is easier on the gearbox then).