Your reply only works if the article were consistently talking about a strict order. However, it is not. It explicitly introduces linear order using reflexivity and antisymmetry, in other words, a non-strict `<=`-style relation, in which equality IS a real case.

If the author wanted to describe a 'no ties' scenario where every object has its own unique place, they should have defined a strict total order.

They may know everything about mathematics for all I care. I am critiquing what I am reading, not the author's knowledge.

Edit: for anyone wanting a basic example, ["aa", "aa", "ab"] under the usual lexicographic <=. All elements are comparable, so "every object has its place depending on every other object." It also "leaves no room for ambiguity in terms of which element comes before which": aa = aa < ab. Linear order means everything is comparable, not that there are no ties. By claiming "no ties are permitted" while defining the order as a reflexive, antisymmetric relation, the author is mixing a strict-order intuition into a non-strict-order definition.