The early McElroy device is a metal block that screws to the pendulum and holds the finger; this is the "thick" model (about 3/8" or 9 mm wide as measured along the pendulum axis). The dot spring attaches to the early style, mounted vertically.
The dot stabilizer is a device intended to eliminate "split dots" from semi-automatic telegraph keys, or "bugs." This common fault of bugs occurs when the dot contact on the pendulum (or vibrating arm) of a bug wipes its corresponding fixed contact (mounted to a post on the base) when it makes or breaks contact, rather than firmly making or deliberately breaking contact.
The pendulum dot contact must be mounted on a spring to allow the pendulum to freely oscillate, even during the make-break cycles of the contacts. But at the moment the contacts make or break, there is no tension on the spring, so the initial make or break is uncertain and results in a scratchy dot (or a "ffft," as McElroy said in his ads).
By not allowing the spring tension to fall to zero, the make is more solid and the break more abrupt. This is done by the "dot stabilizer," which is simply a metal finger positioned to apply tension to the pendulum dot spring. McElroy added such a device to the 1938 deluxe model. He also sold the dot stabilizer separately for 50 cents, so you may find it on other Mac-Keys, added after the fact by their owners.
The early McElroy device is a metal block that screws to the pendulum and holds the finger; this is the "thick" model (about 3/8" or 9 mm wide as measured along the pendulum axis). The dot spring attaches to the early style, mounted vertically.
The later style has a "thin" block (about 1/8" or 3mm wide). With this, the dot spring is separately mounted to the pendulum in the usual manner.
While the McElroy stabilizers may be the most common and best known, stabilizers do show up on other bugs. I found this one on a 1907 Vibroplex. It looks like a cross between the two McElroy models: the dot spring is mounted in its usual orientation, horizontally, but is attached to the stabilizer. My guess is that this is not a Vibroplex item but rather an aftermarket doo-dad offered to telegraphers by another company (or basement tinkerer).
Many handy hobbyists also made their own dot stabilizers. An article by Noel Bevan GW8IH in "Morsum Magnificat" nr. 16, Summer 1990 (p. 7) gave details on his "bug dot damper." He found it was better in eliminating the scratchiness of the dot contact than the common practice of stuffing a piece of sponge rubber in the contact spring elbow.