If your domain is [0,2] then it is not appropriate to integrate over a delta function with payload at 0. More generally, as you state, an integral containing DiracDelta[x-x0] should not have x0 as an endpoint of the integration range.

A way to perhaps understand this is as follows. There are many ways to approximate a delta "function". In order for your integral to make sense, these all have to give consistent results as you let them approach delta(x). But if you try to integrate e.g. f(x)*delta(x) from 0 to 2, different valid sequences of approximations to delta(x) will give results that differ as you let them approach delta(x). So 0 cannot be used as an endpoint of the integration range.