You're talking about synonyms, related words, and hyponyms: words that share a meaning, hold a related meaning, or that are subcategorical words within a larger category. To illustrate using your example with a search in the Merriam-Webster thesaurus for "see,"
- Watch - word related to see (entry 1)
- Gaze - no mention (but listed as related to watch, verb entry 1)
- Stare - no mention (but listed as related to watch, verb entry 1)
- Glimpse - word related to see (entry 1)
Notice how the editors classify two of the words as related, and how the other two words are at least related to watch. This signals that they aren't strict synonyms (there's a denotative difference between seeing and watching) but they are related as words referring to sight-related actions. Here's how Merriam-Webster explains related words:
... this thesaurus includes lists of related words, which are words whose meanings are close enough to the synonymy group to be of interest to the user. These related words do not qualify as synonyms because they have meanings that differ from the basic meaning shared by the synonymy group in some significant way.
The tricky thing is that other thesauruses will list some of these words as synonyms. The distinction is inexact.
If the messiness of how to categorize a word as synonym or related word is too imprecise, another way to describe all of this is to say that gaze, glimpse, watch, and stare are hyponyms of the general verb to see. This is the example that ThoughtCo uses to explain the concept of hyponyms:
Hyponymy is not restricted to nouns. The verb to see, for example, has several hyponyms—glimpse, stare, gaze, ogle, and so on. Edward Finnegan points out that although "hyponymy is found in all languages, the concepts that have words in hyponymic relationships vary from one language to the next"
(Language: Its Structure and Use, 2008).