Does this mean I'm not a 'true' software developer at heart?
Absolutely not. It just means that you like doing things other than coding. There's nothing wrong with that, and it doesn't make you any better or worse than anyone else, developer or not.
Is it possible to become a good software developer without doing extra outside your job?
Of course. Some of that depends on what your work environment is like. If you've got a boring, soul-sucking, job that sucks up all your time and then some doing boring maintenance work, then being able to use work time to develop your skills will be very limited. If you have the kind of environment where you have some latitude to do what you want to explore and learn, then take advantage of that opportunity to develop new skills that also help your employer or co-workers out.
If you find something interesting that you want to learn more about and don't have the time or possibility to use it at work, then you have to weigh that against the other demands on your time, be they family, friends, or hobbies.