What does the "Type 1 query" model on Miles, Sahai, and Zhandry's paper?
Based on the author's presentation at CRYPTO 2016 (available on youtube), we see that the Type 1 query is really modeling the attacker's ability of multiplying the zero-testing parameter by the values it has.

The point is that this abstract query is conservative in the sense that it gives to the attacker more power than it would have (a priori) in practical in multilinear maps setting. Also, this query is modeled like that because it is likely that if an attacker can get the expressions I described as the expected ones, than she/he can also get the ones returned by the queries (for instance, by recovering $g$ and dividing the value $ p_k $ she/he has to get $ p_k / g $).