I don't think it implied anything about Heraclius' views towards the Roman Empire. Romanness was a very flexible concept and the use of the Greek language by the educated was a very Roman thing to do - there is no reason as far as I know for the Romans to think that they were no longer Romans. Greek had been used in official contexts before and the infrequent use of basileus was also attested to in the sixth century. Likewise, Latin continued to be used in the empire, particularly in ruling its western provinces and in the coinage. I went into a bit more detail about this here. This change in titulature is attested to in Heraclius' edicts after 629, but I don't think they represent a universal pattern, especially when the papyrological evidence from Egypt does not attest to 'basileus' at all (the first one being from 641). Hellenised forms of 'caesar' and 'augustus' were still used, likewise for the consulship (though it ended under Constans II, 641-668). It is true that 'basileus' took prominence over them, but it definitely did not replace traditional imperial titles. One interesting suggestion I've read about is that Heraclius was trying to ensure that his eldest son, Constantine III, was higher up in the imperial hierarchy than his younger sons by only giving him the title 'basileus', even though his sons were all supposed to be co-emperors together.

So why did 'basileus' emerge in 629? I don't think we can know for sure, especially as it was a gradual process over many decades and emperors after Heraclius continued to refine the imperial titulature; it doesn't help that our source material for the Roman Empire in the seventh century is very limited compared to other periods. The theory I'm mostly interested in the idea that Heraclius was drawing upon the Old Testament and trying to evoke a comparison with King David. Comparisons with David were common during his reign and 'basileus' was the title of the Old Testament kings in Greek. Within the increasingly Biblicised context of his time, it made sense for Heraclius to go further in an age of unprecedented crisis, with other famous innovations including the use of 'May God help the Romans' in Latin on his coinage and the restitution of the Holy Cross to Jerusalem, which mirrored both David and Christ's entrance to the city. After his surprising victory against the Persians in 628, what better way for Heraclius to reinforce that message than to make the same comparison clear in official documents? This coincided with a reform in coinage (with coins now featuring Heraclius holding a sword and a cross), the forced conversions of the Jews and an intensifying attempt to reunify the various schismatic Churches. Rather than being a turning point between a 'Roman' and 'Greek' empire, I therefore see it as evidence of the increasingly politicised role of Christianity, though obviously imperial rhetoric had long been Christianised and Heraclius was not creating things out of thin air. Anyway, as I pointed out above this was not a universal change nor was it a consistent one, so take this with a grain of salt :)

I think we should first clear up some things about the title itself. Basileus (which is just the Greek word for "King") had been used by Greek-speaking Romans as a title for their rulers for centuries before Heraclius. In the fourth century, Eusebius of Caesaria had used it in his history (where he applied it retroactively for all Roman rulers as early as Julius Caesar), Procopius and Agathias had used it in the sixth, as had most Greek writers, in-between. In fact, John the Lydian, a palace official in the early sixth century who wrote a book on "Roman magistracies", begins his work by discussing what are, exactly, the differences between a rex, a Caesar, and a Basileus.

It is true though that, technically, the Emperors avoided using this title in their official documents. Technically. What they did use, however, were words like vasileia (to refer to their reigns), vasileuein (to rule), vasiliki philanthropia ( imperial charity – a quality good emperors are supposed to possess) etc. (see, for example, the decrees of Justin II, Tiberius II, and Maurice in Jus Graeco-Romanum Vol.III). In other words, every possible derivative except for the actual word itself.

What I’m trying to say, with all the above, is that the use of the title “Basileus” wasn’t some radical, out of left field change that Heraclius just sprang on his unsuspecting subjects. He was just making formal what had been the norm in everyday usage for centuries. Why that Emperor and that particular time to get rid of the older nomenclature – which had been preserved more by cultural inertia than because it had any actual importance by then – is something modern historians have different theories about.

The other answer already mentioned the one about the evocation of King David. Another, by an older historian, J.B. Bury, is that it had something to do with Heraclius’ “conquest” of Persia. He argues that as long as there was another monarch with the title Basileus (which is what the Romans called the Persian Shahanshahs) the Emperors avoided using that title themselves; by now taking that title, Heraclius signified that the Persian king was “reduced to the condition of a dependent vassal.”

Personally, I don’t find either of these two theories particularly convincing. But I’m not particularly knowledgeable about this period as to go more in depth on the subject. One more thing to keep in mind, though, is that Heraclius didn’t just replace one word with the other, but completely overhauled and simplified how the Emperors introduced themselves in their law-decrees. Compare, for example, his decree from 612 with that from 629 (where he first uses the title basileus – again from Jus Greco-Romanum Vol.III):

612 – In the name of the Lord Jesus Christ, our God, Imperator Caesar Flavius Heraclius, faithful in Christ, most serene, greatest benefactor, peaceful, alamanicus, gothicus, francicus, germanicus, anticus, alanicus, vandalicus, africanus, erulicus, gepedicus, pius, fortunate, glorious, victorious, triumphant, ever Augustus….

629 – In the name of the Lord Jesus Christ, our God, Heraclius and Heraclius the New Constantine, faithful in Christ Basileis...