An atlas is a sheaf on the site of cartesian spaces (the site with objects R ), such that ...

One can certainly define smooth manifolds in such terms.

The cartesian site has finite-dimensional real vector spaces as objects, smooth maps between them as morphisms, and is equipped with the Grothendieck topology of jointly surjective families of submersions.

The category of sheaves of sets on this site will be henceforth referred to as the category of smooth sets.

A morphism of smooth sets is etale if its base change
along any morphism from a representable sheaf
is an etale morphism.

An atlas of a smooth set X is an etale epimorphism U→X
whose source is a coproduct of representables.

The category of smooth manifolds admits an obvious
fully faithful embedding into the category of smooth sets.
Its essential image coincides with smooth sets
that admit an atlas.

isthe locally ringed space formalism. He seems to imply it when he says "Daniel likes the option of replacing (1) by the logically equivalent". You can also represent manifolds as sheaves on the site of cartesian spaces, but that is very different. $\endgroup$