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.