Theories can be "wrong", if their predictions contradict experimental
results within the area they claim to be applicable.
Using your terminology, in the case of Flat Earth theory, it does not contradict everyday life experience (taken as experiments). So, it is not "wrong" in this sense, considering everyday life as the area where it is applicable. Of course, in a more precise area, e.g., satellite engineering, it clearly contradicts experiments. So, it is "wrong".
A "mistake" is an action that leads you away from your goals.
According to this definition, if the goal of someone is to put a satellite in orbit, then the Flat Earth assumption is a "mistake" since it prevents the goal.
If you imply somehow that the assumptions of Lorentz invariance or
postulates of quantum mechanics are wrong
Using this terminology, the claim of "two mistakes in physics" is not intended to mean that Lorentz invariance and the postulates of quantum mechanics are "wrong", using your definition of "wrong". Indeed, they agree with experiments. The "mistakes" are conceptual, in the sense that the mathematical realization of spacetime and quantum amplitude as a continuum manifold and a complex number, respectively, are rather restrictive. The mathematical realization of spacetime and quantum amplitude as a causal graph and a combination of multiway graph and branchial space, respectively, is more general. Hence, in these formalisms (Wolfram Model), there are fewer restrictions in order to search for a theory of fundamental physics.
It may turn out that the principles of QM or GR are correct (see
string theory)
I agree, in string theory, there is a proof that quantum mechanics and general relativity are not mutually exclusive, since they can be developed in the same framework. Indeed, S. Wolfram wrote:
I suspect that the continuum limit of the operations I discuss on
character strings is actually related to string theory in the modern
physics sense
In the framework of the Wolfram Physics Project, string theory is a triumph of theoretical physics, despite keeping the "two mistakes" mentioned above, not because of it. One of the promises of the Wolfram Model is to reconstruct the results of string theory as a limit case of a discrete framework. If someday the mainstream approach to physics is able to solve all the problems of fundamental physics, which according to Ed Witten are finite in number (see Physics and Geometry), then there will be no reason to consider that the conceptual structure of mainstream physics contains mistakes. Nevertheless, it may happen that the only way to solve these problems is by overcoming the "two mistakes" mentioned above. Therefore, the status of "mistakes" of the two claims is relative to the Wolfram Model, i.e., if the Wolfram Model can be used as an accurate description of nature, then the "two mistakes" mentioned above are actually mistakes.