“In the most strict sense in the primitive form”. From the historical text, we know the ancients distinguish them, and knew full well during semeai, those with a “larger eye” can kill those with a “smaller eye” (大眼可贏小眼)
From ancient texts like “棋訣” in its section explaining with comments about “大眼可贏小眼”, author and famous Go player (Liu Zhongfu 劉仲甫) in the 11th century, explained what the older text “大眼可贏小眼” meant (the older text 棋经十三篇, we had no idea how old but was already ancient text by the 11th century, and people referenced it and similar contents in the oldest known surviving scroll, and older texts dated at least in the Han dynasty, 2nd century BC to 2nd century BCE).
Liu Zhongfu commented : “局面棋道相交之地，以棋圍之曰眼。不可攻曰實眼，可攻曰虛眼。凡圍棋有雙眼則活，孤眼則死。若黑白相圍，各無雙眼，則大眼贏小眼也。地廣而路多曰大眼，地狹而路少曰小眼。”
And the last sentence is exactly what I said in the previous replies, where the “open area/territory” (地) is larger, it is referred to as a “big eye” (for layman’s terms), and if the open area is smaller, it is referred as a “small eye”. And the phrase he used to refer to as a group that can be killed with one eye is “孤眼”, and the way to kill it is “透點” (poke out the point 點, and not using the word eye). People in ancient times with a more primitive form of counting, they had separate concepts for “eyes” and for a “stricter sense” of “point eyes”. (if they are the same, then he didn’t have to make extra comments just to clear it up, and like today, I imagine even ancient players at the time of the 11th century got “confused” as well)
The way I always understood it is that an eye “is” a single intersection, but as with everything else in Go, if your opponent cannot prevent you from building it even if they get the first move, then you already have it. In that sense a “large eye” is an eye the same way that a bamboo joint is a connection, even though the stones are not literally connected. And a group with four empty spaces in a row has two eyes, even if it’s technically only one large space.
But there is a difference about ko threats. There are no ko threats in what I call an eye, but there are ko threats in bamboo joints or in a linear 4 spaces. A bamboo joint may suffer of lack of liberties making impossible to connect.
Well the thing is you are not going to show the case of ko threats or aji in fights around a bamboo joint or an eye in an eye to beginners, to avoid some more complexity. There is a kind of abuse of language which is fully understandable in the teaching point of view.
There is a difference for how many stones the opponents can “throw in” (how many you can ignore) regarding the end of the game, and relevant to the OP’s question, and the ancient already has a concept for it.
the type of “snake eye space” where the opponent can throw in as many as the space allows, but you don’t have to respond at all (if paired with another 1 intersection true eye, which a snake eye space can easily turn into), and all these “space” are yours in the end regardless you spend moves to capture them (which did matter for the ancient rules using stone scoring).
the type of eye space where the opponent can play as many as they want but you only have to respond once, this is the second tier eye space, where there is only one response needed, and gives rise to lots of confusion, where at which stage does this response have to come in (if it is immediately, or several moves before a response is finally needed?)
the type of “space”, where you have to respond more than once, then it is less like an “eye” but more like territory, where your opponents has the chance to make a living inside if you ignore more than once, or more than twice, or more.
There is a spectrum between a secure “eye”, a big “eye space” of various kinds, toward less and less secure “territory”. Hence I feel it is probably better to refer to the concept of “eyes” more as a property associated with a connected group than trying to define it. Categorizing them makes understanding its property easier. And your skill actually does matter in determine how “secure” (your responses needed") a group really is (the OP’s question)