I'm trying to understand, what an "organization" on GitHub actually is -- an independent account with all capabilities of a user account and additionally some specials or just a "repos container", that need a separate user account for user specific activities. So my question is:
Is it possible to push code to GitHub as organization (using the organization's login) or does an organization always need a personal account to get code into an organization's repository using personal account login?
An organization is only a group of user (contributors) that contribute to one or more projects.
You always commit/push code as a single user, never as an organization (or group) unless everyone share same contributor account (why you would ever do such thing?): that way, no matter who commit/push, for github is the same user (always a user, never an organization).
Moreover remember that commit/push are git concept whereas organizations are github one