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I like Lennart's comment so, Hiro, I won't try to say much more. Stability in the first sense was certainly well understood by the very early 1950's, and it would be hard to be certain of a ``first'' reference. The second sense had to come later since cofiber sequences and fiber sequences were only formalized in 1958 (Puppe) and 1960 (Nomura); the comparison map $\Sigma Ff \longrightarrow Cf$ relating them came a bit later, I believe. Both senses were thoroughly understood before 1964 (when Boardman and I got our theses).