There was always a slight difference between the Roman empire in the east and in the west.

• In the east, they recruited from the local population from the start, often conveying citizenship on recruitment, while in the west recruitment remained limited to people with the Roman citizenship for a longer time.
• In the east, the army was garrisoned in or around the great fortress cities of Syria and environs. (Except for the troops that were in Egypt for a time.) In the west, the army stayed in fortified camps much farther from any centres of civilisation.
• In the east as in the west, the official military language was Latin, but recruits were much more likely to speak Greek among themselves as the local lingua franca, and Greek can be seen more often in surviving military documents from Egypt.
• In the east, there was a greater emphasis on the use of cavalry, archers and slingers, as a counter to the predominantly cavalry-based Parthian (and later the Sassanids) they faced as their primary enemy. Innovations such as the use of heavy lancers and Roman cataphracts started there.

Other than that though, the armies were pretty much the same. There were legions and auxiliary cohorts and alae in the east and in the west.

This remains the case throughout the history of the empire. Our last exhaustive document about the Roman military, the notitia dignitatum or "list of offices" includes amongst other things a long list of Roman military units and is considered to be up to date for around the year 420 in the West, and 390 in the east and paints a similar picture with similar types of units still. (Whether the notitia can be relied upon to provide an accurate picture if a far more fraught question.)

However, by this time (Late fourth/early fifth century) BOTH the Eastern and Western armies had stopped resembling the legions you may know from film or video games.

At this time, the army was divided in several large field armies (5 in the east, 7 in the west according to the notitia) called Comitatenses and garrisoned troops called limitanei. The old-style legions and auxiliaries could still be found in the limitanei but as much smaller and less important formations. The old 5000 man strong legions no longer existed, instead everything seems to have been divided in units 600-1000 men strong.

In the field armies (and also the garrison forces) we see many new unit names. For example: auxilia palatina or auxiliaries of the palace, which had a higher status and importance than legions lacking the palatina designation. Units called vexillatio (originally a detachment from a legion, now a word for permanent units of cavalry). Imperial guard regiments are called scholae instead of praetorians (which had been disbanded). Units called numeri of which we're not sure what they were. (it's a vague label that in previous centuries was used for less formally incorporated allied forces.)

The soldiers themselves looked and fought differently. There was heavily armoured cataphract and clibinarius cavalry, horse archers, and lancers. (Clibinarius is probably derived from a baking oven, referring to how hot soldiers and horses clad from head to toe in mail armour got, which if true makes it my favourite name for an ancient soldier.)

Infantry wore trousers, carried long spatha swords and a variety of stabbing or throwing spears as well as throwing darts called plumbatae, they used oval or round shields (not square scutum style) and wore mail or scale armour. (not the famous segmented plate armour of the early empire.) They seem to have fought in tight-knit shield walls, instead of the loose individual fighting style of the Republican and early imperial legions.

So, now for the million sesterces question: When and how did the Roman army change?

This is the frustrating part. We don't know.

At the start of the third century we see some growing emphasis on stuff like armoured cavalry, but on the whole the army of Septimius Severus looks pretty familiar to the one we know from the time of Trajan. And then the Crisis of the Third Century happens. Between all the civil wars, invasions, plagues and disintegration of the empire, our sources also dry up. When the soldier-emperors put the empire back together in the late third/early fourth century, what we see emerging from the chaos is quite different than what went in to it. But when did these changes happen? And why? Was it as early as Galienus, or as late as Constantine? Was the army split like this to provide a strategic reserve, or to prevent usurpations? There is little we can say for certain.

The changes were probably gradual. We already see the legions being split in smaller and smaller units in the Principate, in practice if not formally, because it became too hard to just uproot entire legions wholesale, and easier to just split off smaller detachments. And various emperors probably raised various new units and gave them various new names and cool titles to keep them loyal, and then they stuck around. But here we are reduced to careful speculation.

Frankly, it would have been more surprising if the Roman army had not changed. The army of Augustus was different from the army Scipio used to fight Hannibal. The army of Constantine was different from that of Augustus. And this change did not stop: if the 4th century field armies can still be recognised in the armies of Maurice (an Eastern Roman emperor from the late sixth century) they can't really be seen in the armies of emperor Nikephoros Phokas II in the 10th. (I note these emperors because they both have military manuals attributed to them describing their armies.)

So in summary: Yes, the Roman army in the east was pretty similar to that in the west, even if there were differences in emphasis. Changes that happened happened everywhere in the empire, until the western half disintegrated in the 5th century. But changes did happen both in east and west, and the classical Roman army changed somewhere around the 3rd/4th century.

(The "rotating every six minutes" thing sounds like some kind of myth though. There are some vague references in our sources to Roman maniples being rotated in and out of combat, and much ink has been spilled on discussing how this might have worked in practice, or on arguing that it never happened at all, but if one thing is certain it is that nothing so definite as "rotating every six minutes" can possibly be said.)