Register a loop with my phone

There are several apps you can use. I personally use Locus Map for that purpose. You should download a map separately from an external provider. I use Openandromaps.

Other apps that do the trick are OSMAnd (comes with its own maps) or Oruxmaps.

You could have a look at the OSM wiki and browse for an app you like.

Note that in any case you have to add the new loop manually to OSM as you need to check for any existing overlaps and connection to the road network. But different editors help by allowing you to draw from the recorded track.