The Following is my communications with Olgur Celikbas in researchgate:
Ogler:You may want to check: $R=k[[x,y]]/(x^2, xy)$ and $M=R/(x)$
me:does still such an emaple exist if we suppose that R is unmixed of dimension 2?(Unmixed = every associated prime is of height zero)
Ogler:Try this one: the ring of the example is from a paper of Costa, Huneke and Miller, titled "complete local domains of type 2 are CM".
$R=k[[x,y,z]]/(xz,yz)$. Then $dim(R)=2$, $depth(R)=1$ ($x-z$ is a non zero-divisor) and $R$ is reduced (so that associated primes are height zero.)
Let $M=R/(z)$. Then $depth(M)=2$, i.e., $M$ is MCM.
It looks like $Ext^0(k,M)=Ext^1(k,M)=0$ and $Ext^2(k,M)=k$ so $Type(M)=1$.
(Type means the vector space dim of the first non zero $Ext$ w.r.t $R/m$)