Yup that pretty much covers it, though with older guns I think everything with that action was called Model 10/110, before they got into 12, 14 etc for specific models. So sometimes you will hear about the 10/110 meaning the action in that generic sense, like you do with the Rem 700.
You can also tell a bit from some of the letter codes used after the number if you're a geek like me - F is synthetic (FC for accustock), B is laminate, L is lefty. V is varmint (heavy barrel), SS is stainless. So 16FCSS tells you its short action synthetic accustock stainless even if you didn't know the 16 is the weather warrior.
WRT to the cheap ones, the Stevens still uses the 10/110 action but with a cheaper stock and without the nice accutrigger etc - they can be a good starting point for custom builds if you plan to replace the stock and trigger anyway, as its a cheap way of getting a good action. The Axis is a totally different budget action though, I haven't used one but they have a reputation for not being very nice.
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