There is just the circle of fifths. It can be run in two directions. For everyone who, like you, insists that the *direction of dominance* is the "circle of fourths" there is at least one who insists that it is the "circle of fifths", so this “distinction” is only a source of confusion.
Usually, when speaking of, e.g., a “circle-of fifths” bridge, we almost always mean “Rhythm bridge”, i.e., in its simplest form (in the key of Bb): D7-G7-C7-F7. It’s very common, so a first step may be learning to identify it and improvise over it. Then, of course, there is the distinction between “breaking out and circling in” (as in I Got Rhythm) or “circling out (and breaking in” as in the bridge of Topsy:
(Bb7-Eb7-Ab7-Db7/F7 in the key of bbm. Exploring these is certain to keep you busy before you even contemplate tritone substitutions or introducing ii-V pairs or superimposing, e.g., a Bb major or blues scale over a Bb Rhythm bridge.
To confuse matters further, there is also the “diatonic” circle of fifths, as in Fly Me to the Moon (in C: am7, dm7, G7, Cmaj7, Fmaj7, bm7-5, E7 …, note the tritone step that keeps all the bassnotes within the C major scale).
And zillions of variations.
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