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No but yes but no.
I’m not sure you are serious/genuine here but I’m also not sure you deserve the downvotes and I have some time to kill.
“Republic” is “rule by the people” (Rex public). It means the same as democracy (“people rule”). The only difference is Latin vs Greek.
As used in the US, your description kinda-sorta covers the principles that the parties once maybe kinda stood for 150 years ago or so. (Something like “more federal power / less federal power”).
In reality these days they’re just labels for two different groups, and the words have no connection in this context, it’s just an historical note.
I was talking about meaning in entire world, not american local dialect.
That’s fine.
In that case you can feel free to ignore the US-specific portion and go back to the first “no, they are synonyms” statement.