You are correct!
The term Pirouette Je Matoosh is not correct!
The video below is an example of a Fouetté Pirouette.
“Fouetté” is French for “whipped,” and it’s that whiplash motion of the raised leg that gives the turn its brilliance. That quick up-and-down of the ballerina’s foot sometimes sends her traveling downstage; but the ideal sequence of 32 fouettés never travels from a single spot.
Men can do fouetté turns, too. But part of the excitement with a ballerina is that she does most of the turn on point. She comes off point as her raised “working” leg shoots out to the front and sweeps around to the side; but then, as she rises up onto point, that working leg whips inward.