Just a piece of information from the french side. Dieudonn'e, in ``La g'om'etrie des groupes classiques'' (Springer, 1970), takes the definition of a classical group for granted. But browsing through the table of contents, it's clear he means $GL_n(K),SL_n(K),O_n(K,f),U_n(K,f),Sp_{2n}(K)$ plus variants (e.g. the projectivized versions).
Alain Valette
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