Greg wrote:

Take a look at the opening paragraphs of De Ente et Essentia. In particular:

6. … the word “essence” must signify something common to all natures, by means of which (nature) diverse beings are placed into diverse genera and species; as, for example, humanity is the essence of man, and so with other things.

7. And because that by which a real thing is constituted in its proper genus or species is what is signified by the definition expressing what the real thing is, philosophers sometimes use the word “quiddity” for the word “essence.” This is what the Philosopher often calls what something was to be, i.e., that by which it belongs to something to be what it is.

8. It is also called form, in the sense in which the word “form” signifies the full determination of each real thing, as Ibn-Sînâ says in the second book of his Metaphysics.

9. Further, it is given another name, nature, taking the word “nature” in the first of the four ways given by Boethius in his book On the Two Natures. In this way, whatever can in any way be grasped by the intellect is called a nature. For a real thing is not intelligible except through its definition and essence.

10. The Philosopher, too, says in the fifth book of the Metaphysics that every substance is a nature. But the word “nature” taken in this way appears to signify the essence of a real thing according as it has an ordering to the thing’s proper operation; and no real thing lacks a proper operation.

11. The name “quiddity,” however, is taken from the fact that what is signified by the definition is the essence. But it is called essence from the fact that through it and in it a real being has existence.

...

14. In composed substances there are form and matter, for example, in man soul and body.

15. But we cannot say that either one of them alone may be said to be the essence. …

16. Neither can the form alone of a composed substance be said to be its essence, although some try to assert this. For it is evident from what has been said that essence is what is signified by the definition of a real thing. And the definition of natural substances contains not only form, but matter as well; otherwise natural definitions and mathematical ones would not differ.

17. Neither can it be said that matter is placed in the definition of a natural substance as something added to its essence or as something outside its essence, because this mode of definition is proper to accidents, which do not have a perfect essence. This is why accidents must include in their definition a subject which is outside their genus. It is clear therefore that essence includes matter and form.

18. Further, neither can it be said that essence signifies some relation between matter and form or something added to them, because this would of necessity be an accident or something extraneous to the real thing, and the real thing would not be known through it. And these are traits of essence. For through the form, which is the actuality of matter, matter becomes something actual and something individual. Whence what supervenes does not confer on matter actual existence simply, but such an actual existence; as accidents in fact do. Whiteness, for example, makes something actually white. Whence the acquisition of such a form is not called generation simply, but generation in a certain respect. It remains, therefore, that the word “essence” in composed substances signifies that which is composed of matter and form.

What I gather from this is that form is that which determines a being to be the kind of thing it is while essence is just what it is. Therefore essence would be the true bearer of being's quiddity.

RomanJoe wrote:

In other words, what Socrates is (his essence) is different than his substantial form because form is merely a determining principle of his matter and not a definitional principle of his being. Is this correct?

“. . . a definitional principle of his being” is a dark phrase.

RomanJoe wrote:

What I gather from this is that form is that which determines a being to be the kind of thing it is while essence is just what it is. Therefore essence would be the true bearer of being's quiddity.

Aquinas seems to say, in no. 11, that essence and quiddity signify the same thing, which he would deny of form.

(When Aristotle introduces the notion of formal cause in Physics II.3, he introduces it just as the "formula [logos] of the essence". As this notion is put to work in Aristotle's philosophy it is being refined, I think.)