1. lit. A deck covering half the length of a ship or boat, fore or aft: in this sense still used in some small partly open craft.
a. In old ships of war: A deck extending from the mainmast aftward, situated between the then smaller quarter-deck and the upper or main deck. After the two decks above the main deck were reduced to one, for which the name ‘quarter-deck’ was retained, ‘half-deck’ survived only in the expression ‘under the half-deck’, applied to the part of the main deck from the main mast aftward, formerly covered by the ‘half-deck’.
b. In colliers: A deck under the main deck, extending forward to near the after-hatch and containing berths, etc., for the crew (obs.). OED
It's definitely not referring to b., because none had more than one deck. None of the ships were war ships, so I doubt it would be definition a.. Definition 1. seems the most likely given the small size of these ships.
Half-decked is a common term used to describe the construction of boats. It was first used in the early seventeenth century and is still in use today.
A whole deck, on the other hand, has a deck covering the entire length of the ship, i.e., not open-topped.