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).
In the book Groupes de Lie classiques''(Hermann, 1986), R. Mneimn\'e and F. Testard define classical Lie groups in their introduction: same list as in Dieudonn\'e, but assuming of course $K=\mathbb{R}$ or $\mathbb{C}$.
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).