Sauce Messine should fall into the bechamel category. Velouté sauces incorporate stock rather then cream or milk.
This is a basic answer to the difference in the two, from YIANNISLUCACOS .
The difference between velouté and béchamel is that the milk is replaced by a stock, i.e. chicken, beef, fish etc.
In every other respect velouté sauce is almost identical to béchamel.
In general, it is used as a base for a number of white sauces. Its main applications are:
- Base for sauces
- Base for soups
- Basic ingredient for mixtures and fillings, as it offers moisture and a rich texture, such as patties, pies, pasta fillings.
As for other sauces with parsley, watercress, or other ingredients, they would fall into certain categories based on the mother sauce base, not the flavors or other additions.
It is a fairly easy internet search to get more information, should you need it.