The first is played by the codified rules of classical chess see Tactical TeamChess.
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As a more precise use of the term ‘Cooperative Chess’ refers to chess games played by two teams (each having more than 1 player) with classical set of chess pieces (2*16) on a classical chess board (8*8) see Tactical TeamChess.
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Cooperative Chess (or CoopChess, Co-op Chess) is a synonym of Tactical TeamChess.
As a more precise use of the term ‘Multiplayer Chess’ refers to chess games played by two teams (each having more than 1 player) with classical set of chess pieces (2*16) on a classical chess board (8*8) see Tactical TeamChess
Otherwise the expression of "Real Teamchess" is a synonym of Tactical Team Chess.