I know that an ordered set is one which satisfies transitivity and at least one of the following hold for all $a, b$ $$a<b, a=b, b<a$$ And for a totally ordered set 'at least one' is replaced by 'exactly one'
My question is whether an ordered field is a field with an order or with a total order?
An ordered field is a field with a total order. Note, however, that it's not enough for the field to be equipped with a total order; that order has to interact nicely with the field structure. E.g. if $a<b$, then we need $a+c<b+c$ for all $c$, and there is no reason for this to be true of a general total order on the field.
Incidentally, your terminology here seems odd. Usually (as far as I'm aware) the term "ordered set" refers to a totally ordered set. What you call an "ordered set" is a (linearly) preordered set. I wonder if you meant "at most one" instead of "at least one"; if so, then what you call an "order" is more commonly called a "partial order".