The sentinelhub Python package, as described on the Github page, allows users to make OGC (WMS and WCS) web requests to download and process satellite images within Python scripts. It is like using the API, but using Python code.
On the other hand, eo-learn is a collection of open source Python packages allowing a user to develop a data-analysis workflow that includes different tasks, such as cloud masking, image co-registration, feature extraction, classification, etc… For example, a user could use eo-learn to predict land-use based on machine-learning techniques (example here) or determine water levels of a water body (dam, reservoir, lake, …) from satellite imagery (example here). You can find a more detailed overall explanation on the package in the introduction page of the package.
My aim is to get the data using the API without using any source from my machine.
In your case, the sentinelhub-py package is the most adapted. You can follow the examples shown in the documentation here as a starting point, then adapt the code to fit your specific needs.