Coming into harbour, masting was usually struck down to foot and topmast level.
The top-gallant mast, and the Royal, the next step up, were never struck until well out to sea. They catch the faster winds up higher, the ship can easily be pushed onto her beam ends. This can make a mess when still in harbour, with no hull speed for stability.
You're close to right Rosie. The sails are named for the mast and section of mast that carries them.
Coarse sail, lower-top, upper-top, top-gallant, royal.
It is the fifth one up, way up.
The main mast was equal to deck length at the top of the top-mast.
Then stepped above that comes, the top-gallant-mast, then the royal.
Fore-mast, Main-mast, Mizzen-mast, all stepped to the Royal
Any square sail higher than that Royal, and cookie is drying his tea-towells.
Edit.
Scanned from Vol 1 of BR 67 (2).
Sorry about the scan quality at the bottom of the page.
If Master and Commander, is of British origins, this is Royal Navy, Admiralty Pattern. They way I was taught.
It may help.
This shows one only top-sail, but, many ships were split top-sailers, with an upper and a lower top-sail.
And the term
ship, actually strictly applies, only to a three masted vessel, fully square-sail rigged on all three masts.
All others are schooners, ketches, brigantines, barquentines, clippers or some other type of sailing vessel.