I'm curious... what is meant by "pivot" that is different in other dances? I've never heard the term swivel, I don't think. (maybe in a swing class?) I have a vague memory of hearing pivot in ballroom, but I can't remember how it was different from a tango pivot.
For the eyes of the curious ONLY - and in answer to a direct question ....
A (Natural) Pivot: By definition, "A turn to the right on the ball of the LF holding the RF in front in contrary body movement position".
Pivots are usually danced by the man, stepping backwards, while lady dances the counterpart, a Pivoting Action (A turn to the right on the ball of the RF, the LF NOT held in CBMP). The turn is executed on the ball of foot, but the feet are not closed, but apart.
There is no real basis for comparison with AT, as AT does not (to the best of the writer's knowledge) have a concept of contrary body movement (from which the related foot position is derived). To a casual observer, a series of repeated pivots can look as though man and lady are rocking backwards and forwards, with their feet apart, turning about a 1/2 turn on each step, and thus moving along the line of dance, taking it in turns to take forward and backward steps, each time turning around each other, on a spot created by the closely placed feet between them. That's not an accurate description of what is happening, mechanically, but that's what it looks like.
A 'swivel' is like an AT forward step with pivot, the turn is on the balls of feet, with the feet together. In AT (to the best of the writer's knowledge), this action can be danced forwards or backwards. The defined tango technique of at least one teaching society makes the equivalence explicit between a 'ballroom' swivel and a tango 'pivot', and my saying so will now probably be the cue for someone, predictably, to post that the teaching societies should mind their own business etc. If they do, don't expect me to respond. The bait doesn't taste agreeable.
It doesn't surprise me in the least that you can get around a ballroom floor, as a tango follower, because you have nearly all the tools to succeed. Lead & follow is the essence of ballroom dancing (but most of the people who try it, can't do it very well), but it remains of the essence. If you establish a connection with someone who can lead, you'll be amazed at what might happen next (just like tango

). Unfortunately, if your main interest is in Viennese Waltz, your opportunities to 'pivot' are very limited, because the standardised figures do not use the action. However, outside the constraints of international style competition (where only the standardised figures are permissable), it is perfectly acceptable to dance a series of natural pivots, and their use in the 123 123 rhythm create a new rhythm 12 31 23, which is great fun to try.