Not quite. Plenty of eighteenth-century topsail schooners (as Americans would call them; the British would simply call them schooners) didn't set a topgallant or royal - or even a course - on the foremast. Sometimes, as the term implies, just a topsail. And an eighteenth-century (or early nineteenth-century) schooner might well carry topsails and topgallants on both masts. For an example, see the famous Sultana (square sails on both masts) or the Prince de Neufchatel (full set of sails on the foremast, and one on the mainmast).
I really think the big distinction - in most places and in most periods - is whether there are gaff-rigged sails on both masts. If yes, it's a schooner (or, if there are square sails on the foremast, and if you're talking to an American, topsail schooner. If no, it's a brig (if there are square sails on both masts), a snow (if the boom-rigged mainsail is set to a "snow mast," just aft of the mainmast itself) or a brigantine (if there are no square sails on the mainmast).
But, as I noted earlier, different contemporary sources contain different definitions - and British sources differ from American ones. (The British, it seems, generally don't use the term "topsail schooner"; some American sources do and some don't.) And when we get into non-English-speaking countries, the topic becomes pretty hopeless.
The good news is that usage of the word "schooner" isn't as complicated as that of the word "sloop." The permutations of that one, and the distinction between "sloop" and "cutter," go all over the place. (There's at least one case in which the British Navy reclassified a vessel from "cutter" to "sloop" because the commanding officer got promoted from lieutenant to commander. And then there were the American revenue cutters, nearly all of which were rigged as topsail schooners. Heaven help us.)
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