Wikipedia's “topographical_summittopographical summit” article says:
In topography, a summit is a point on a surface that is higher in elevation than all points immediately adjacent to it. Mathematically, a summit is a local maximum in elevation. The topographic terms "acme", "apex", "peak", and "zenith" are synonyms.
Apparently “synonyms” in that last sentence means “synonyms of summit”. However, the article goes on to use the concept of topographic prominence (“also known as autonomous height, relative height, shoulder drop ... or prime factor”) to differentiate summit from its synonyms:
The term "summit" is generally only used for a mountain peak with some significant amount of topographic prominence (height above the lowest point en route to the nearest higher peak) or topographic isolation (distance from the nearest point of higher elevation)
and then gives a few examples and lists UIAA (a climbing association) criteria used to designate independent summits.
Regarding multiple summits on one mountain, the topographic prominence article uses Everest's South Summit (ca. 8749 m altitude) and main summit (ca. 8850 m altitude) as an example in explaining why K2 (ca. 8611 m altitude) rather than South Summit is the world's second highest mountain:
While Mount Everest's South Summit ... is taller [than K2], it is a subsummit of the main summit. Only summits with a sufficient degree of prominence are regarded as independent mountains.