I can do that.
We'll start with the obvious: they're both Constitutional. That is, the structure of the government is dictated by an established Constitution, which we obviously have. Fair enough, move on.
A democracy is a type of government in which the power to make change lies with the people. It is in contrast to an autocracy (where the power lies with one person), a theocracy (the power lies with the church), and lots of others.
A republic is a type of government in which power is spread across many people. It literally means "public thing" and is in contrast to a monarchy, where all of the power resides with the monarch, and every executive, legislative, or judicial proceding is done by them, or in their name.
Because it would be impossible for everyone in our country to vote for every issue, we have a representative democracy, in which each area elects a representative to go to the capital (DC) and represent our area's wishes in the federal government. These days, practically every government that is a republic is also a representative democracy and vice versa, so in practice the two terms are basically interchangeable. You are obviously correct in that we are definitely a republic, but we are also a representative democracy.
None of this is partisan, or opinion, or a secret; you should have all learned it in Middle or High School, so if you need a source, you can dig out your old textbook, and it's in there. Like any science, the terminology in political science is determined by the scientific community; otherwise, they couldn't communicate to advance the field. You can redefine something if you like, and there's no impenetrable rulebook to say you're wrong, but if you're not using the same terms as everyone else, you're probably just going to end up shouting into the void.
So yeah, our system as designed is both a republic and a representative democracy.