LEF will concentrate on the remise and redoublement, pretty much as a single category of action. Surely the line of the renewed attack is not sufficiently important to force a change of definition.
Note: In modern fencing, redoublements may include all aspects such as compound and simple attacks, feints, changes of angle, changes of type of attack, actions on the blade, etc. Is any of this visible in the period texts? It is not a part of them or is it assumed from foil/epee training?