To look for a time before the word "Shinto" was used, you must look back to the Asuka period, as the question suggests. At this time, the educated term for Japanese ritual was jingi 神祗. But I would caution against thinking of this as Shinto. The formation of Shinto as a nationwide ideal for ritual, practice, and teaching, did not happen in the classical period, even though there were already ritual traditions at Ise, Izumo, and other places. The Kojiki and Nihon Shoki, at that time, were simply history books, not sacred texts, and the kami were part of the Buddhist macrocosm.
We should instead think of jingi as the word for Japanese court ritual in contrast to Buddhist ritual. This contrast faded into the background with the end of the Heian period. In the medieval period there was shinbutsu-shūgō 神仏習合, free mixture of kami and buddhas, and identification of kami as avatars for boddhisatvas, and thus little need for a separate category. I think modern Shinto really began with Yoshida Shintō, a medieval attempt to construct a non-Buddhist theory of shrines, and developed apace in the Edo period.
In the Edo period, literate Japanese said that their country had "three teachings" 三教. This was a borrowing from Chinese civilization, where Confucianism, Buddhism, and Daoism existed alongside each other, sometimes in conflict and sometimes in harmony. In fact, back at the dawn of the Heian period the Japanese Buddhist genius Kūkai had written a book called Sangō Shiiki 『三教指帰』 that discussed these "three teachings"; the third was Daoism, even though there were no Daoists in Japan (that we know of), because he was endeavoring to compare it to Buddhism. Kūkai was responsible for putting the Japanese kami in a Buddhist cosmos.
But by the Edo period, the third teaching was no longer Daoism; it was already Shinto, or kami-no-michi. Shinto was recognized as having its own ritual tradition and style of practice, which was clearly not Chinese. This was due to the medieval tradition of commentary on the Nihon Shoki and Yoshida Shintō, as well as other pre-Edo ritual practices which are not so well-understood.
What happened to Edo period Shinto? You could write a book about this -- indeed, I plan to -- but to put it very simply, the Kokugaku school claimed that the existing schools were too private in character; that is to say, they were too religious. The goal of Kokugaku was to create a "national teaching", but simultaneously to create a public-facing teaching, something like what we would think of as "secular".
So what did ancient Japanese people call their animism? Likely nothing at all, since there was nothing to contrast it with.