Strictly speaking, an everywhere non-negative prior that integrated to some finite positive value other than 1 would not be a proper density and so arguably could in that sense be referred to as "improper", but since
(a) if it integrates to $k$, say, it's easy enough to scale if you need that
(b) frequently in Bayesian work we're only dealing with forms up to normalizing constants anyway, such as: (i) because of the need to integrate the denominator in Bayes' theorem so we'd just be applying a different scaling constant to that calculation (e.g. if we recognize the form of the posterior we can work out the right normalizing constant directly), or (ii) the need for explicit normalizing is removed by other considerations (such as if we're generating via accept-reject, say, we don't necessarily need the explicit normalizing constant)
So a prior for which we failed to give the normalizing constant to make it integrate to 1 doesn't usually pose any problem since we can get the result as if we had done so.
As a result, generally when people say "improper" they literally mean one with a non-finite integral, for which no "correct" normalizing constant exists