Will AES-GCM smaller sized tag (mac) be included in larger sized tag I want to make sure that I understand AES-GCM correctly:

If I have:
key1, plaintext1, iv1, adata1 and requset tag size of 96 bits, and tag size of 128 bits, will the first 12 bytes of the 128 bits tag equal 96 bits tag?
Yes, that's what the standard mandates.

First we need to note that if you fix a key and an IV, then AES-GCM becomes fully deterministic, that is the same input will yield the same exact output.

Then we look into the standard, NIST SP800-38D (PDF), and look at the way the tag is generated.

$$T=\operatorname{MSB}_t(\operatorname{GCTR}_K(J_0,S))$$

which effectively tells us that the $t$ most significant bits of the tag output (which is the same for both invocations) are to be used for the tag.

Then we look up the semantics of the $\operatorname{MSB}_s(X)$ function which are:


  The bit string consisting of the $s$ left-most bits of the bit string $X$.