I think the confusion stems from how specific software name their procedures for doing specific types of analysis. In your case, perhaps stems the confusion from SPSS that has the repeated measurements ANOVA and mixed ANOVA procedures, which one allows only for within subjects covariates, and the other for both within and between subjects covariates (mixed).
In any case, you have a repeated measurements design in which you measure the same outcome over time for the subjects. Logically, for the mean of your outcome you’d want to include the Time and Group effects, and possibly also their interaction.
To obtain correct/efficient inferences, you’d want also to account for the fact the measurements on the same subject are correlated. You can do that using different types of models. For example, marginal models which directly postulate a specific structure for these correlations (e.g., compound symmetry (= sphericity assumption), AR1, continuous AR1, Gaussian correlation, etc.).
Or you could also use a mixed model, in which you specify random effects (the fixed effects correspond to what I previously mentioned over the mean of your outcome). To make it clear, what you do in the random effects directly translates to assumptions for your correlations. There you typically start from random intercepts (that corresponds to compound symmetry), and you can also include random slopes, etc. You can statistically test which model gives you a good fit in the correlations.