Ballroom dancing rotation (turn) is accomplished in several different ways. In general, Latin/Rhythm dancers' rotation happens between the floor and the ball of the foot (SWIVEL TURNS). The Standard/Smooth dancers, in general, rotate through the muscles of the leg from the foot to the bottom of the hip (LEG TURNS).
The term "TURN" is used in ballroom dancing to describe rotation. The term "SPIN" is not generally used. The phrase "Spin Turn" in Standard/Smooth dancing is the name of an element, not a description of the turning action. Generally speaking, rotations of over 60 degrees are executed as "Swivel Turns" on the balls of the feet. Rotations of 60 degrees or less are executed as "Leg Turns" through the muscles of the legs. Some rotations are executed as a combination of both "Swivel Turns" and "Leg Turns" over multiple steps.
Ballroom rotations are executed as solo two-legged turns, assisted two-legged turns and four-legged turns. The four-legged turns are almost always executed in "closed" dance position. Almost all of the Latin/Rhythm rotations are solo and assisted two-legged turns. Almost all of the Standard/Smooth rotations are four-legged turns. In the two-legged turns, the rotation happens on the supporting leg of the dancer. The four-legged turns are executed simultaneously on the supporting leg of each partner.
Let's take a look at some Latin/Rhythm rotation examples. The "Under Arm Turn" element in Rumba , Cha Cha, etc. is an assisted two-legged "Swivel Turn" that can be powered or un-powered. The "Walk Around Turn" element is a solo two-legged "Swivel Turn". An exception to the rule is the "Back Spot Turn" element. This element is a four-legged rotation, danced in closed dance position implementing "Leg Turns". The "Toe Heel Swivel" in Swing for example is a two-legged turn of less than 60 degrees but is still danced using the "Swivel Turn". Turning on the ball of the foot allows for very quick turns that can be powered by one or both partners. This is what makes double turns possible.
Let's look at some Standard/Smooth rotation examples. The "Open Left Turn" element in Foxtrot and Waltz, etc. is a turn of 120 degrees, danced in closed dance position, executed over three steps. This is a turn of less than 60 degrees on each step which is the definition of a "Leg Turn". The "Open Side Locks" pattern involves two simultaneous solo three step turns. The first two steps are "Leg Turns" and the third step is a "Swivel Turn". All of the elements of Viennese Waltz are danced with "Leg Turns. There are no "Swivel Turns". The "pivot" element in Standard/Smooth dancing is a rotation of more than 60 degrees on each step. By definition the "Swivel Turn" is danced on each step of this movement. The "Spin Turn" element of Waltz, Foxtrot and Quickstep is really a partial pivot. The rotation is less than 60 degrees but because it is a pivot, the "Swivel Turn" is used. Turning through the muscles of the leg allows the dancers to turn very smoothly and under very precise control.
"Swivel Turns" are executed by having a slight bend in the knee of the supporting leg. That puts the weight on the ball of the foot. It is very difficult to swivel with weight on the heel of the foot. The leg and body needs to be toned to such an extent that all of the rotation happens on the ball of the foot and none of the rotation is in the muscles of the leg or body. This takes a while for the dancer to learn to feel how toned he/she should be. Sticky floors require more, slippery floors requires less, etc.
"Leg Turns" are executed by allowing the body to rotate through the current supporting leg. The dancers don't turn by using the muscles of the leg, the dancer allows the leg to rotate.
I have used the phrases "in general" and "almost always" quite a bit in the previous descriptions. There are always exceptions in ballroom dancing. Very seldom are there rules that have no exceptions. Most of the specific numbers I use are a middle of the road specification. These are all variable based on the skill of the dancer, his/her flexibility, length of the dancers legs, partnership, etc. Once you understand how it works mechanically, repetition will increase your flexibility and make it be an automatic action.