How to determine the filter type based on the impulse response?
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.