It's almost impossible to say , since the Syriac and Armenian Christian sources cited above are far earlier than any of the Islamic material, which all comes from a century later. It's almost impossible to know what these people were thinking or where they were getting a lot of their information from, and in how faithful of a form that it was being written down. Like I said above regarding tafsir, some scholars like Gabriel Said Reynolds have even made claims that the exegetes of the Qur'an were in some cases creating exegetical fiction of aspects of Muhammad's prophetic career (which is of course a very different claim than 'Muhammad didn't exist' - which I don't believe).
The picture gets even more problematic when we read Syriac texts such as the Chronicle of Khuzistan, written in the 660s. The relevant portion of the Chronicle says this:
>In the city of Istakhr, they made Yazdgard from the royal lineage the king. With [Yazdgard] the kingdom of the Persians [would] end. He set out and came to Mahuzē and appointed a general named Rustam. Then God brought against them the sons of Ishmael, [who were as numerous] as sand upon the seashore. Their leader was Muhammad. Neither walls nor gates nor armor nor shield withstood them. They took control of the entire Persian Empire.
The Arab conquest of Persia began in 633 and lasted until 654 AD. Muhammad could not have bee the leader of the Arabs invading Persia, because the invasion did not begin until a year after his death. At the very least, it seems like Muhammad was alive a year or two longer than Muslim sources say.
And to cast even further doubt on the exact nature of early Islam, we know that Muʻāwiya, the first Umayyad caliph who reigned 661-680 AD, did stuff like the following according to the Maronite Chronicle, a Syriac Christian source from the mid to late 7th century A.D. -
>In the year 971 [660/61 c.e.], the eighteenth of Constans, many Arabs assembled in Jerusalem and made Muʻāwiya king. He ascended and sat at Golgotha. He prayed there, went to Gethsemane, descended to the tomb of the blessed Mary, and prayed there.
Why is this caliph, if he is anything like modern Muslims, praying at Golgotha and Gethsemane if Muslims were said to believe that Jesus was not crucified? Why did the coins of Muʻāwiya's reign all have the cross on them? The Maronite Chronicle also says the following:
>He struck both gold and silver [coinage], but it was not accepted because it did not have a cross on it.
This is somewhat of an enigmatic reference in the scholarship. Who did not accept the coins? One interpretation is that Christians in the area did not accept the coins without the cross, the other interpretation is that it is of course Muʻāwiya himself who did not accept cross-less coins, because all Umayyad coins from this era have the cross depicted on them.
It's all so weird, because early non-Christian historical references, while sometimes filled with slander against Christians ('the Eucharist is cannibalism!'), all seem to support the basic historical facts about Christianity - i.e. that Christians were followers of a man who was crucified and came back to life and refused to worship pagan idols. We don't have any of this nonsense about Muhammad being alive when he should have been dead, Muhammad seemingly invading multiple places at the same time, numerous references to the early Muslims following the Torah, or Muslim caliphs praying at Gethsemane and Golgotha and sending out coins with Christian imagery on it for decades afterwards. And all of this is really the tip of the iceberg.