@Eric So, first things first: you don't have diagrams only commutative up to homotopy. You have coherent commutative diagrams: the homotopies making the diagram commute are part of the datum, and so are the higher homotopies ensuring those homotopies are compatible, and so are the even higher homotopies etc.

But granted that, yes there is a similar property. The correct notion of "first order thickening" here is "square zero extension". This basically turns the theorem into the definition of étale though...

Indeed the cotangent complex is exactly the obstruction to lift against square zero extensions

The definition of square zero extension is in HAGII.1.2.1.7

Using the cotangent complex as obstruction to lift against square zero extensions is the topic of HAGII.1.4.2, in particular 1.4.2.5