I'm having a hard time figuring out how exactly cross validation works in practice:

To do K-fold cross validation on a data set, you divide your data into K sets. Then for each fold $i$, $1 \leq i \leq K$, you fit a model $M_i$, for which you get an out of sample error $MSE_i$.

Now you have a cross validation error:

$CV_K = \frac{1}{K}\sum{MSE_i} $

You repeat this process for different sets of hyper-parameters, chosen using grid search, or Bayesian optimization or some suitable method, and go with the set of hyper-parameters that give you the lowest $CV_K$.

So far it is clear.

My understanding is: the models $M_i$ will have the same hyper-parameters, but they won't have the same fitted parameters (coefficients in a linear model, weights in a neural network, etc...).

So which model from the $M_i$ models do you actually go with?

Or is it the case that once you have chosen you hyper-parameters using CV, you then refit a new model using the entire test data set? If this is the case, isn't there a chance that the new model performs worse than all of the $M_i$.

I'm having a hard time figuring out how exactly cross validation works in practice:

To do K-fold cross validation on a data set, you divide your data into K sets. Then for each fold $i$, $1 \leq i \leq K$, you fit a model $M_i$, for which you get an out of sample error $MSE_i$.

Now you have a cross validation error:

$CV = \frac{1}{K}\sum{MSE_i} $

You repeat this process for different sets of hyper-parameters, chosen using grid search, or Bayesian optimization, or some other suitable method, and go with the set of hyper-parameters that give you the lowest $CV$.

So far it is clear.

My understanding is: the models $M_i$ will have the same hyper-parameters, but they won't have the same fitted parameters (coefficients in a linear model, weights in a neural network, etc...).

So which model from the $M_i$ models do you actually go with?

Or is it the case that once you have chosen you hyper-parameters using CV, you then refit a new model using the entire test data set? If this is the case, isn't there a chance that the new model performs worse than all of the $M_i$.