This is almost 100% correct, except that a "coat of arms" strictly applies to a garment with the escutcheon or armorial achievement embroidered on it, (whereas most people think of a full achievement when they talk about a "coat of arms.") This is all laid down in statue from the time of Henry VIII.Yes a coat of arms is strictly speaking only for an individual not even a family, I think, and they don't necessarily persist unchanged down through the generations. If that person married someone who also had a coat of arms then the two would be merged based on all sorts of arcane rules and the next generation would have a modified one. Or at least that's how I understood it. However these days I'm not sure that gets done much any longer.
In addition, the use of "Family Crests" (the part of the overall coat of arms that is a a three-dimensional object at the top) was probably a Victorian invention - they also invented Clan Tartans. Having a coat of arms or family crest is quite rare, and having the hereditary right to use it is even rarer. The many companies advertising on the internet to sell you merchandise with your supposed “family crest” on it are usually not engaged in legitimate genealogical research, and have more likely made up a crest that may not belong to your family at all.