Gorenstein dimension vs projective dimension

Gorenstein dimension is a kind of homological invariants of finitely generated modules.

Are the following results true?:

(1). Let $R$ be a Noetherian local ring and $M$ be a finitely generated $R$-module with finite projective dimension. If $Ext_{R}^{1} (M,R) =0$, then $M$ is projective.

(2). Let $R$ be a Noetherian local ring with unit and $M$ be a finitely generated $R$-module with finite Gorenstein dimension. If $Ext_{R}^{1} (M,R) =0$, then $M$ is projective.

Solution 1:

(1): let $R$ be a regular local ring of dimension at least $2$, and $M=k$ be the residual field (not projective). Then $M$ has finite projective dimension (because regularity by https://stacks.math.columbia.edu/tag/065U, eg Prop 10.110.1), and $\mathrm{Ext}^1_R(M,R)=0$ because $R$ has depth at least $2$ (by https://stacks.math.columbia.edu/tag/00LE, Prop 10.72.5, then https://stacks.math.columbia.edu/tag/00NN, Lemma 10.106.3).

(2): I expect the same kind of counter-example, but I can’t find a good reference for Gorenstein dimension in Stacks.