Question is as mentioned in the title:

Are there any introductory notes on deformation theory that are easier to read for differential geometers?

I am learning about differential graded Lie algebras (and $L_\infty$-algebras). I am aware of Marco Manetti's notes Deformation theory via differential graded Lie algebras.

Are there any other notes that are easier to read for people who are trained in differential geometry?

I know some commutative algebra, for example definition (and few properties) of Noetherian, Artinian rings. I can recall (or read) some more commutative algebra if required.

I read from the paper From Lie Theory to Deformation Theory and Quantization that

"Deformation Theory is a natural generalization of Lie Theory, from Lie groups and their linearization, Lie algebras, to differential graded Lie algebras and their higher order deformations, quantum groups."

So, this gave some hope that there would be some notes from the point of view of Lie theory/differential geometry.