The same four signs are called several things in different centuries, and a reader will meet all of them.
Tropical is the Greek term and the oldest, from the Sun's turning at the solstices. Movable is the usual medieval Latin rendering, and it shifts the emphasis from the astronomy to the character: these are the signs that move. Cardinal is the modern English standard, from the Latin cardo, a hinge, and carries the sense of a pivot on which a thing turns. Initiating and creating are looser modern descriptions rather than technical names.
None of these is wrong and they are not quite synonyms. Tropical is a statement about the Sun's path, and it also names the tropical zodiac itself, the one measured from the equinox rather than from the fixed stars, which is a different and much larger distinction that this entry does not settle. Movable and cardinal are statements about how the signs behave in a reading.
The confusion worth flagging is a different one. The three sign qualities, cardinal, fixed and mutable, are called the modalities in almost all modern astrological writing, while the three house groups, angular, succedent and cadent, are called modalities in some of it. The atlas keeps them apart by using quality for the signs and modality for the houses. That is not universal usage, and it is stated here rather than left implied, because it is the only way to name both groups without ambiguity.
Sources TetrabiblosClaudius Ptolemy with Christian AstrologyWilliam Lilly