The 22nd Amendment was passed after Truman was elected, so it did not apply to him. (ex post facto)
He was eligible but chose not to run.
You’re right, I’m in a fog but the other side is that many presidents chose not to run for a third term, FDR broke the tradition of not doing so, had Truman run again it would have been essentially running for a third term since FDR died so early in his last term.