How to determine the filter type based on the impulse response? When given a discrete impulse response, let us say it is from an FIR filter so it is finite, how can I assume the filter type?
The task is to recognize or to preclude one or more of the following types:


Low pass
High pass
Band stop
Band pass


For example, I read that a zero at $h(n = 0)$ (where h is the impulse response of my filter) means that the filter is a low pass but how can I say that with precluding the band stop immediately?
If you know for sure that the filter type is one of the four given ones, then the exercise is quite straightforward. Look at the zeros of the corresponding transfer function

$$H(z)=\sum_{n=0}^{N-1}h[n]z^{-n}\tag{1}$$

where $N$ is the filter length. The zeros at DC ($z=1$) and at Nyquist ($z=-1$) are given by

$$H(1)=\sum_{n=0}^{N-1}h[n]\tag{2}$$

and

$$H(-1)=\sum_{n=0}^{N-1}h[n](-1)^n\tag{3}$$

It should be obvious that a (classic) low pass has a zero at Nyquist but no zero at DC. For a high pass filter the opposite is the case. A band pass filter has zeros at DC and at Nyquist, whereas a band stop filter has no zeros at DC or at Nyquist.

PS: What you read about the zero of the impulse response is either non-sense, or you misunderstood what they meant.