When one speaks of the version of Orthros in the parish as being "incomplete" - they are comparing it to the official text as it appears in the Horologion of the Church, and as is described for in older versions of the Typikon. But it should be said that, at least in the practice of the Patriarchate of Constantinople's parishes, most parishes use a truncated Orthros (a version of the orthros with the modifications made by the Patriarchate in the 19th century). The only places that you will find the "full" Orthros in a Greek-practice church will be in a monastery, although not all do the "full" version. What that book is missing are the following:
1. Two psalms that fall before the opening Trisagion (in fact, the prayer "O Heavenly King" does not belong there)... I think they are Ps. 19 + 20... Nowadays, these psalms will be heard at the beginning of Orthros during Holy Week (when they are done at night, typically).
2. The reading of 2 Kathismas of the Psalter: one before the hymns of the 1st Kathisma, one between the hymns of the 1st and the 2nd kathismas. This seems to have fallen 100% out of practice in the parishes in the Greek tradition.
3. The Chanting of the Full Canon (8 Odes numbered 1-9 skipping #2) after the Anavathmoi - this would involve 14 hymns per ode, combined resurrectional/day themed and for the saints. Also seems to be 100% out of practice.
4. The chanting of the full set of psalms (140-143?) with "Let every breath." Instead, the 1st line of the 1st psalm "Praise him..." is chanted, and then the final verses of the 2nd and 3rd psalms are intersperced between hymns.
5. The whole ending of the matins: after the chanting/reading of the doxology (depending on the magnitude of the day; sundays it is always a chanted great doxology), there is supposed to be the Completion litany, the Aposticha, Trisagion, Apolitikion (which is preserved in the modern usage as "today Salvation has come" on Sunday, and the Apolitikion of the day on weekdays), Fervent Litany, and Dismissal. Instead, much of this is done silently by the clergy during the chanting of the doxology. (As an aside, I do not believe the Aposticha would be chanted on Sundays anyway, but it doesn't matter- that whole part has been cut out.)
I hope this was helpful!