is a town with under 700 people a village?

No - it's more complicated than that - in the UK, anyway.

In the UK, the main historical distinction between a hamlet and a village was that the latter had a church, and so usually was the centre of worship for an ecclesiastical parish.
However, some civil parishes may contain more than one village. The typical village had a pub or inn, shops, and a blacksmith. But many of these facilities are now gone, and many villages are dormitories for commuters.
The population of such settlements ranges from a few hundred people to around five thousand. A village is distinguished from a town in that:

* A village should not have a regular agricultural market, although today such markets are uncommon even in settlements which clearly are towns.
* A village does not have a town hall nor a mayor.
* If a village is the principal settlement of a civil parish, then any administrative body that administers it at parish level should be called a parish council or parish meeting, and not a town council or city council. * However, some civil parishes have no functioning parish, town, or city council nor a functioning parish meeting. In Wales, where the equivalent of an English civil parish is called a Community, the body that administers it is called a Community Council. However, larger councils may elect to call themselves town councils.
* There should be a clear green belt or open fields, as, for example, seen on aerial maps for Ouston surrounding its parish borders. However this may not be applicable to urbanised villages: although these may not considered to be villages, they are often widely referred to as being so; an example of this is Horsforth in Leeds.

Yeah, lots, like Springer Town in Oaklahoma