Properties of orders and preference relations Suppose I have alternatives $A$, $B$, and $C$. If I have strict preferences, that means that for any $x,y \in \{A,B,C\}$ such that $x \ne y$, either $x \succ y$ or $y \succ x$. Assume transitivity, non-reflexivity, completeness.

Things get a lot less clear to me when I'm trying to talk about it as an order. How would I say that an agent has an order over the alternatives that would allow some ties? And how would I say an agent has an order over the alternatives that would not allow ties? Is it just the intuitive weak total order and strict total order, over alternatives where the weak and strict carry the same meaning as for preference relations and total is a synonym for complete?
You are right, total is a synonym for complete, it simply means that two elements can always be compared.

Regarding your other point, I don't think that your definitions of strict order and weak order are common but they make perfect sense. Notice, however, that they differ from some definitions used by mathematicians and therefore you have to specify what you are talking about to avoid possible confusions (you can check for instance "strict weak ordering").