There are multiple ways to formalize the notion of a (limit) sketch, which are basically equivalent. This makes it a bit difficult to decide on a "right way" to formalize sketches. One nice property would be that a category of models (in say $\mathsf{Set}$) is given by a unique theory up to equivalence. 


My (probably flawed) understanding is that based on a particular weak definition of "sketch" this need to be the case and that a sketch may care about the way we axiomatize our theory, like e.g. whether we use a constant and a binary operation to say what a monoid is or a sequence of operators $(\prod : A^n \to A)$ or something else.


Let's only focus on limit sketches (since these yield the theories I actually care about). 



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> What is the correct notion of limit sketch where category of models are given by a unique theory up to equivalence?
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A notion of limit sketches that makes theories unique up to equivalence
If I understand correctly your question you are looking for some definition of limit-sketches such that if two sketches $\mathcal T$ and $\mathcal T'$ are Morita-equivalent (that is the categories of their $\mathbf{Set}$-valued models are equivalent) then they are equivalent as categories.


If that is the case they you are doomed to fail. The problem lies down to the fact that any reasonable definition of limit-sketch should comprise algebraic theories as a special case, meaning that any algebraic (i.e. Lawvere) theory should be a limit-sketch.


Unfortunately it is well know that there are non equivalent algebraic theories giving rise to the same categories of algebras (models). 


As an example, taken from *Adàmek, Rosicky and Vitale's Algebraic theory*, you can consider the algebraic theories $\mathcal N$, the full sub-category of $\mathbf{Set}$ spanned by the finite sets $[n]=\{1,\dots,n\}$, and its full subcategory $\mathcal T\_2$ spanned by sets of the form $[2n]$. In the above mentioned book is proven that the category of models $\text{Alg}(\mathcal N)$ and $\text{Alg}(\mathcal T\_2)$ are both equivalent to $\mathbf{Set}$ but that two theories are not equivalent as categories.


