In Europe (at least continental), there is a clear distinction between "village" and "town", because historically, "town" was a legal status granted by the king or church (city would be perhaps the town where the archbishop would have his seat), but it wasn't necessarily distinguished so much by its size, they'd be all very small.
Personally, I'd but the line between "town" and "city" to having some kind of rapid rail transit. (That'd mean, for example, that in the country where I live is just one city.)
My point was that the size of a 'small town' varies by context, and that a town of 1,000 would be tiny even in the context of a place that doesn't have many people.
Also, 'city' in the UK is to do with a special government/crown-granted status, not really the population per se.
Personally, I'd but the line between "town" and "city" to having some kind of rapid rail transit. (That'd mean, for example, that in the country where I live is just one city.)