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When switching between ML-solutions, the title line should change (saying something like “Maximum Likelihood Estimate (best)”, with the “best” being changed for the other solutions. You can also manually choose the solution by selecting “Estimation -> Select Estimate”; if only one ML solution is shown there (after giving the model some time to find potential alternatives) or if the shown ones have the same parameter values, all’s fine, and you can work with the results.
> What would it mean for my sample if Onyx tells me it’s overspecified only based on the sample?
potentially nothing; as the test for overspecification is numerical only, it may simply be wrong. If you get different solutions with the same fit value, then you may want to investigate what the differences are (e.g., if you find that the loadings of one factor seem arbitrary and you find the factor in fact has almost no variance, then you can conclude that the indicators seem to have no reliable common factor).