It can be confusing to many what the BBT is, especially if Inflation theory is forced into it, though it is an elegant model to solve two important issues.
BBT was born from Lemaitre's Primeval Atom theory. Lemaitre's published a teaser in 1925, followed by a full GR paper in 1927 that was the first GR solution that fit observations. Contrary to all other models, Lemaitre's model explained the universe as an expanding one, whereas Einstein, de Sitter and others were stuck on the premise that the universe was static, never really changing.
Lemaitre's model assumed the universe to have begun at an unknown size, but he didn't care too much for guessing this since the bulk of his modeling explained the universe from today to such a tiny point. Indeed, there are no observations of the universe when it was tiny for reasons predicted by his model. Namely, the universe would have been comprised of elementary particles in a sea of photons and these particles, especially electrons.
The "Bang" moniker came in 1949 from a rival theorists, Frey Hoyle on a radio show.
Inflation was not introduced until 1979 by Alan Guth.
No doubt many scientists are comfortable with adding Inflation theory to the BBT, but a caveat now and then is warranted.
It is also customary to give Hubble credit for discovering that "distant galaxies appeared to be receding away from each other, and the further apart they were, the faster they were going. In other words, the universe isn't static; it is expanding."
Hubble made it clear that galactic velocities were "apparent velocities". In de Sitter's GR model, he predicted such things, though their motions would be through space, not of space carrying those galaxies away from one another. Hubble never accepted the expansion model, though in his last years in retirement he suggested it could be the case.
The first to discover that the "apparent velocities" were a result of the expansion of space, of course, was Lemaitre (1927). His paper calculated the first linear expansion rate for the universe in this paper. [Lemaitre used Hubble's galaxy distances and Slipher's redshift velocities.] Hubble's more robust work was not till 1929. He held tightly to the credit afforded him and Mt. Wilson for their work.