There is the algebraic definition of Morita equivalence:
Two algebras A and B are called Morita equivalent if the categories of left modules $A\text{-Mod}$ and $B\text{-Mod}$ are equivalent as abelian categories.
Now I want to consider this in a categorical context, so I want to compare two monoidal categories. I came across two different types of Morita equivalence:
1) In here (and in different sources) they define weak Morita equivalence as follows:
Let $\mathcal{C}, \mathcal{D}$ be tensor categories. They are called weakly Morita equivalent if there exists an indecomposable right module category $\mathcal{M}$ over $\mathcal{C}$ such that $$\mathrm{Fun}_{C}(\mathcal{M}, \mathcal{M}) \cong \mathcal{D}$$
2) in Etingof's "Tensor categories" they define categorical Morita equivalence:
Let $\mathcal{C}, \mathcal{D}$ be tensor categories. We will say that $\mathcal{C}$ and $\mathcal{D}$ are categorically Morita equivalent if there is an exact $\mathcal{C}$-module category $\mathcal{M}$ and a tensor equivalence $\mathcal{D}_{op} \cong \mathcal{C}_\mathcal{M}^* := \mathrm{Fun}_\mathcal{C}(\mathcal{M},\mathcal{M})$.
What is the difference? Is there a connection between those two? I always used the first one, but the name suggests it is a weaker thing then the second one. Is there a more common one?
In my work I start with the algebraic version and then switch to the categorical one. I want to give an example of two categories and check if they are Morita equivalent. I checked the first one, but now I got unsecure if that is enough.
Thanks in advance!