The issue is Thesis 2. The notion of a canonical inner model is vague, but I don't think that when inner model theorists use the term they mean to suggest that the model is obtained by iterating a sharp. In common usage, the canonical inner models include every proper class Mitchell-Steel premouse whose countable $\Sigma_1$-elementary substructures are $(\omega_1+1)$-iterable mice. (Yes, you can iterate the measurables into weird configurations or take stationary tower ultrapowers in forcing extensions, but you can't mess with the internal theory, which is all one really cares about.) It is known that these models can be close to $V$ – actually equal to $V$ – in the presence of many Woodin cardinals.

The Inner Model Problem ("Build a canonical inner model with a supercompact cardinal") is vague. If one knew the general definition of the term "canonical inner model," one would have answered half the question. The vagueness of the Inner Model Problem is not an issue because in actuality the problem has more than one component:

1. A pattern that is known to hold quite high into the large cardinal hierarchy with many applications.
2. The vague problem of extending this pattern to larger cardinals.
3. A host of precise yes-or-no test questions that likely can only be answered positively by solving (2) and to which a negative answer would strongly suggest that (2) cannot be solved.

These test questions include the HOD Conjecture, the Ultimate $L$ Conjecture, Con(Supercompact + $(\Sigma^2_1)^{\text{Hom}_\infty}$ wellorder of the reals), Con(Supercompact + Ultrapower Axiom), $\text{AD}^++V = L(P(\mathbb R))$ implies GCH in $\text{HOD}$, and more!

The issue is Thesis 2. The notion of a canonical inner model is vague, but I don't think that when inner model theorists use the term they mean to suggest that the model is obtained by iterating a sharp. In common usage, the canonical inner models include every proper class Mitchell-Steel premouse whose countable $\Sigma_1$-elementary substructures are $(\omega_1+1)$-iterable mice. (Yes, you can iterate the measurables into weird configurations or take stationary tower ultrapowers in forcing extensions, but you can't mess with the internal theory, which is all one really cares about.) It is known that these models can be close to $V$ – actually equal to $V$ – in the presence of many Woodin cardinals.

The Inner Model Problem ("Build a canonical inner model with a supercompact cardinal") is vague. If one knew the general definition of the term "canonical inner model," one would have answered half the question. The vagueness of the Inner Model Problem is not an issue because in actuality the problem has more than one component:

1. A pattern that is known to hold quite high into the large cardinal hierarchy with many applications.
2. The vague problem of extending this pattern to larger cardinals.
3. A host of precise yes-or-no test questions that likely can only be answered positively by solving (2) and to which a negative answer would strongly suggest that (2) cannot be solved, or at least that our current conception of a canonical inner model must be drastically altered.

These test questions include the HOD Conjecture, the Ultimate $L$ Conjecture, Con(Supercompact + $(\Sigma^2_1)^{\text{Hom}_\infty}$ wellorder of the reals), Con(Supercompact + Ultrapower Axiom), $\text{AD}^++V = L(P(\mathbb R))$ implies GCH in $\text{HOD}$, and more!