What would be an example of using this method:

public virtual bool SaveEntity(
   IEntity2 entityToSave,
   bool refetchAfterSave,
   IPredicateExpression updateRestriction,
   bool recurse
);

or this one?

public virtual bool SaveEntity(
   IEntity2 entityToSave,
   bool refetchAfterSave,
   IPredicateExpression updateRestriction
);

Or why and when would you want to use these?

When you want to restrict the save process in such a way that for example a given field in the entity to update (it only works with updates) has a given value, so you can implement a concurrency scheme the way you like it.

So for example if we have a table called Employee defined as:

EmpId int identity(1,1) Primary Key, FirstName, LastName, Gender

We could write code so that all employees whose name is Bob would have their gender set to 'M'?

Is that what you are saying?

That's functionality provided by UpdateEntitiesDirectly().

SaveEntity() with a restriction is used when you have concurrency rules based on a timestamp column for example, or you only should update a row when the original values aren't changed. You can then construct a filter on these fields and pass it in with the SaveEntity() which then only succeeds if the filter succeeds.