While looking into this question https://tex.stackexchange.com/questions/565031/slightly-bigger-integral-in-exponent I found that the formatting of \int differs slightly when using the amsmath package vs the base LaTeX with the former using a scriptstyle integral while the base uses a textstyle integral.

Given the following equation:

e^{i \int \!dt\,\frac{1}{2}m\dot x^2}

I get 
with amsmath loaded and
without it loaded.

In both cases, the definition of \intop is the same and there doesn’t appear to be any functional difference between the redefined \int in amsmath and the definition of \int in fontmath.ltx

So the questions are: (1) what’s happening differently in amsmath to change the appearance of the integral and (b) which is the correct behavior?