So, the question appeared more subtle than I initially thoughts so I have written a short paper with more references and the details of what I'm going to say below. It is available [here][1].

Here is the summary:

First, Makkai's theorem cited by Ivan is indeed false. Building on Ben Wieland's comment (thank you very much for this!), one get the following counter-example.

Let $I=B(\mathbb{Z}/2\mathbb{Z})$ the category with one object $*$ and one non-trivial automorphism satisfying $f^2=1$  and let $C$ be the subcategory of $I \times \omega$ containing all the objects, and all the arrows $(*,n) \to (*,m)$ when $n < m$ but only the identity when $n = m$.

In particular, functors $I \to C$ are all constant, as $C$ has no non-trivial endomorphisms, but there is a non-trivial functor $I \to Ind(C)$: indeed the colimits of the chain of the $(*,n)$ is in the Ind completion, and as a presheaf, it is the pullback along the projection $C \to I$ of the unique representable presheaf, so it comes with a $\mathbb{Z}_2$-action and hence is a non-trivial functor $ I \to Ind(C)$, which doesn't belong to $Ind(C^I)$.

As $C$ has no non-trivial endomorphisms it is Cauchy complete, hence $C$ identifies with the full subcategory of $\omega$-presentable objects of $Ind(C)$ and hence the exponential $Ind(C)^I$ is a counter-example to Makkai's theorem.


Now this construction can be generalized to any category and any ordinal, and this leads to the following theorem:

**Theorem:** Let $I$ be a category and $\kappa$ be a regular cardinal. The following condition are equivalent.
1) $I$ is $\kappa$-small, has no non-identity endomorphisms, and its posetal relfection is well-founded.
2) for all category $C$, the functor $Ind_\kappa(C^I) \to Ind_\kappa(C)^I$ is an equivalence.
3) for all $\kappa$-accessible category $A$, the category $A^I$ is $\kappa$-accessible with its $\kappa$-presentable objects being the functors from $I$  to $\kappa$-presentable objects of $A$. 


In particular, one recovers Lurie's version of the theorem (I mean for 1-categories), as well as Meyer's result mentioned by Peter Haine. Also Makkai theorem is false even for uncountable $\kappa$. But this isn't the end of the story. If we are interested in category with colimits, then everything works more nicely and we get a positive answer to my questions:

**Theorem:** Let $I$ be a category and $\kappa$ be a regular cardinal. The following conditions are equivalent.
1) $I$ is $\kappa$-small.
2) for all category $C$ with $\kappa$-small colimits, the functor $Ind_\kappa(C^I) \to Ind_\kappa(C)^I$ is an equivalence.
3) for all locally $\kappa$-presentable category $A$, the category $A^I$ is locally $\kappa$-presentable, with its $\kappa$-presentable objects being the functors from $I$  to $\kappa$-presentable objects of $A$.


The proofs of these and additional details are in the paper linked above.


It seems Makkai's theorem was used in a few places in the literature - but from what I can tell, all uses I've seen are covered by these two theorems.

  [1]: https://arxiv.org/abs/2307.06664