How to open this recessed faucet aerator? It seems very hidden inside!
From the original posted picture, it appears that you have the type that needs an aerator wrench.
See link below 
aerator with wrench
To remove the aerator, without the specific wrench, try using a pin wench.  
The aerator is inside the faucet, so you will not be able to use slip grip, etc. on the outside. 
Those two notches are what you have to grap to be able to twist the aerator off.  By luck, a very thin long nose plier may be able to work.
The wrench cost about a $1 at HD or Lowes.

This is where the notches would be located based on original pictures. The notches would be located inside the rim. 


Your image showing location of notches. 
Sorry If I mislead you,  After closer exam of the side view, maybe this in not a notched aeration.  Can you provide a complete picture of the side view of the end of the facuet.

1: https://www.zoro.com/american-standard-aerator-metering-brass-m962393-0020a/i/G0853912/