In the literature, some authors define ``quasi order'' as transitive and reflexive, others define it as transitive and irreflexive.

No such discrepancy seems to exist in using ``preorder'' for the former (transitive and reflexive) and ``strict partial order'' for the latter (transitive and irreflexive).

It seems wise to use only the unambiguous terminology, and start any text where the term ``quasi order'' is felt necessary with a proper warning.

Just for completeness: a partial order is transitive, reflexive and antisymmetric.