“the[y] must have a common part and a distinction, and common part can’t be the distinction”
The statement from post 1 “So each person has two parts, essence and distinction, because essence is not their distinction” is incorrect. The error is in the idea that distinction is a part. God is simple, there are no parts.
There are relations in God. Relation can refer to distinctions that are internal to a substance. St. Thomas Aquinas uses Aristotle’s idea of relation, see Categories). “Relation” means that which makes something of, than, or to some other thing.
Aristotle, Categories: Part 4 - Expressions which are in no way composite signify substance, quantity, quality, relation, place, time, position, state, action, or affection. To sketch my meaning roughly, examples of substance are ‘man’ or ‘the horse’, of quantity, such terms as ‘two cubits long’ or ‘three cubits long’, of quality, such attributes as ‘white’, ‘grammatical’. ‘Double’, ‘half’, ‘greater’, fall under the category of relation; ‘in a the market place’, ‘in the Lyceum’, under that of place; ‘yesterday’, ‘last year’, under that of time. ‘Lying’, ‘sitting’, are terms indicating position, ‘shod’, ‘armed’, state; ‘to lance’, ‘to cauterize’, action; ‘to be lanced’, ‘to be cauterized’, affection.
No one of these terms, in and by itself, involves an affirmation; it is by the combination of such terms that positive or negative statements arise. For every assertion must, as is admitted, be either true or false, whereas expressions which are not in any way composite such as ‘man’, ‘white’, ‘runs’, ‘wins’, cannot be either true or false.
classics.mit.edu/Aristotle/categories.2.2.html