No, he wasn't - at least not in the sense implied by the modern-day term.

First of all, the Twitter user you linked to is trying to argue about terminology on the basis of an English translation of the work in question (Babur's autobiography, the Baburnama). For example, he counters the quote from Stephen Dale that Babur used the term "Tengri" instead of "Allah" by pointing to the word "God" in the English translation, which he insists must stand for "Allah." Likewise, he points to the appearance of "Holy War" in the English translation and insists that this must stand for "Jihad." But the original doesn't say "Jihad," the original word used was "Ghaza." There's something a little odd about building a semantic argument on the basis of a translation of a given work without reference to the original words used. Anyway, that's a minor point.

Historians of the early Mughals and the early Ottomans share a similar problem in figuring out how to grapple with the role of religion in the formation of these two empires. Both came into existence in frontier zones where Muslim and non-Muslim peoples were highly mixed, in the Ottoman case the Turkish-Byzantine and Balkan frontier and in the Mughal case in India. Both states eventually adopted (to varying degrees) a self-identity that made use of Islamic rhetoric. However, it would be naive to accept this rhetoric at face value without recognizing the purpose for which it served: as a particular mode of self-presentation meant to provide the state or the dynasty with legitimacy in the eyes of an Islamic audience.

The term "ghaza" (gaza in Ottoman Turkish) referred to above has been a contentious one in Ottoman history for this very reason. The early Ottoman rulers claimed to be gazis (people who take part in gaza) and this served as a legitimizing tool for them. To Islamic jurists, gaza was a form of holy war, and that's why the translator of the Baburnama chose to depict it that way. But what was gaza, beyond the jurists' definition? The more historians of the Ottomans have examined the way the term was actually used, and what sort of actions were actually depicted as falling under the umbrella of gaza, the less it seems to have matched its technical definition. We find that the Ottomans had no problem incorporating Christians into their army, allying with them, marrying them. Gaza was vaguely Islamic, but the term could also be used largely without the religious undertones. Much of the time, it simply means "raiding" without a religious connotation. If Christians could take part in gaza, going along with the rest of the Ottoman army on raids and conquests to win booty for themselves, was it really "holy war"? To some of the people involved it may have been, but not to everyone. People were free to read into the concept of "gaza" whatever they wanted - it could both be holy war and not be holy war at the same time. And all the while the Ottoman rulers depicted themselves as gazis fighting for Islam.

So what did "ghaza" mean in 16th century India, and what did Babur mean when he said he was "going on campaign for ghaza" (ghazā ishi bilä safar qilib)? He could have meant a lot of different things, and his own personal interpretation may very well have changed over time. According Jos Gommans, Babur's early career was characterized by an open-ended, fairly inclusive and non-scholastic understanding of Ghaza that only began to change late in his life, after his conquest of an empire, into a more self-consciously Islamic depiction (Mughal Warfare, p. 46). Babur's followers in encountering the term could have interpreted it in multiple ways simultaneously, depending on the nature of their own religiosity. The fact that the term had a particular meaning in a scholarly Islamic context doesn't imply that it would have been understood the same way outside of that context.

Thus, the irony of the Twitter-user's statement, "I’m sure that you needn’t be told what “Ghazi” means," when it's actually about as far from clear as it could possibly be. As a historian of the Ottomans, I would certainly love to have someone from the 14th century explain to me what Ghazi means. The Twitter-user is, in effect, reading (a translation of) Babur's text and taking it at face value, as an accurate reflection of how Babur and his followers understood the nature of their military activities. But it would be wrong to make assumptions about their understanding of "Ghaza" (much less "Holy War" in English translation) simply based on the term alone.

Whatever Ghaza meant to these people, it wasn't the same as the image that "Jihadi" brings to mind now in the 21st century. Babur may have used Islamic rhetoric in his autobiography, but that doesn't mean that he used that rhetoric in the same way as modern Jihadists or that the terms and concepts involved would have been understood by him in the same way. Nor did it mean that his actions corresponded to the image presented by such rhetoric. After all, his self-presentation in literary form near the end of his life wouldn't necessarily have directly and accurately reflected his thoughts and behavior at large even at the time of writing, let along decades prior. Babur and his world were complex and very different from our own. Bypassing that complexity to call him a "Jihadi" would reduce him to nothing more than a modern-day political football.