Totalitarianism is widely understood to be a mode of governance in which no individual or collective freedoms are recognised, in the sense of a "freedom" being a mode of behaviour which is explicitly permitted by the government and can be undertaken without punishment by the government. As with nearly all things, totalitarianism should not be viewed as a yes-or-no proposition. One can ask to what degree the Galactic Republic, Plato's Republic, the United States of America, or The Fifth Republic of France are totalitarian — or conversely and more to the point, the extent to which these countries guarantee various freedoms and abide by those guarantees. The answers will be different in each case, and at different epochs as well, because the jurisprudence of each nation changes with time.
Without knowing which freedoms the Galactic Republic guaranteed, it is difficult to say the extent to which it was free from tyranny before Palpatine became Chancellor. However, Episodes I – III, shows evidence of the Republic being an organisation which is so unable to act that its member states can go to war without any effective intervention by the Senate, but which at one point abandons the right to change the Chancellor, and subsequently authorizes actions of war against member states and the Jedi order without significant evidence for the causus bellum (much in contrast to precisely the crisis of Episode I). This strongly suggests that the Republic was initially free from tyranny, only to devolve into totalitarianism at the prompting of a charismatic orator, finally culminating in the dissolution of the Senate by the first thirty minutes of Episode IV.
Aside from this, there is the question which you might raise of what having "one governmental system" has to do with totalitarianism; and whether this condition even holds in the case of the Galactic Republic. The Republic was supposed to govern the entire Galaxy — if you ignore pockets of lawlessness such as Tatooine, that is; it can only be said to have been ambiguously involved in the Republic, if you wish to view rule by the Republic in absolutist terms — but each member state had its own government. (Naboo apparently had some unspecified sort of elected monarchy, for instance.) So, it's clear that the galaxy had more than one "system of government" in the Star Wars mythos; and in any case, this is not pertinent to how free the citizens were under the Republic (where the absence of freedom is the defining notion of totalitarianism).
As in all things, it is essential in order to be able to answer such questions to know what one means by the terms. If you cannot say what precisely a totalitarian state is — or rather, what it means for one state to be more totalitarian than another in some regard — then it is unlikely that
you will be able to arrive at a conclusion. And if you and your friend disagree on what it means for a state to be "totalitarian", this is an opportunity to discuss what priorities you have in choosing your definitions, what qualities are more important to identify, and indeed how you can discover what ideas are true or useful to consider.