The data API is great for searching for imagery and getting the image IDs. When you use the GeometryFilter you will receive all of the images that fall within you AOI, even if only a little bit.

However once you have your image ID, if you want to download just the clipped version of the image you will want to use the Orders API. By placing an order with the clip tool enabled  you will receive only the part of the image that is within your AOI, similar to the subscription API. 

Hi, Octave.

I also have related question when searching for data using API.
You mentioned that I will receive all images that fall within my AOI, even if only a little bit.
So, are there no other options for obtaining products that fully satisfy my AOI?
Because I only want to get the data that fully satisfy my AOI.

Thanks.

Yes, unfortunately as of now the data API does not allow you to filter for area coverage. In explorer those calculations are happening on the fly. You can however calculate the area coverage given the footprint geometry for each scene that is part of the Data API response. Here is a github notebook on how to do that.