Which of the two perceptions is representative of our truth? 

Both play a part in the essentially "true" versions of ourselves. However, the important thing to focus on is that one builds off of the other. Self-awareness is the foundation for how other people perceive you. When you build your character and personality to your own liking, the exact same character shows externally to others.

A stable outwards perception, then, is dependent upon a stable inner perception. You must be content with yourself before others can. Although both matter to some varying degrees, one builds off of the other and that's an important thing to note when deciding which one of the two represent YOU.

Which one is more important?

Since we've established that self-awareness is the foundation for external perception, we can say that it is more important. Perfecting how you appear adds stunning details and adornments to a building, but a strong building must first be built from the beginning with a strong physical foundation. The character you know about yourself is the person you will live with, while others may come and go.

How can I become "the best"? 

Components of success and the workplace require one to compete with other fellow peers. This is an obvious certainty. In order to get the job, you must be better than other applicants. While this is true, it is best to be mildly aware of your competitors but extremely aware of your work and how you can be better/improve. When you have the mindset that you are competing with only yourself, you become more optimistic as a result, and it is likely that your work will be stellar and land you a job. Trying to battle competitors you know almost nothing about is also, very simply, inefficient and not worthwhile.

To really truly perfect your work and be the best at what your doing, you should analyze your own weakness and improve them. Focusing on others is a problem because

a) It can potentially breed malicious intent/ jealousy that would inhibit you from working well alongside future colleagues, and

b) It can cause you to lose motivation and optimism. It promotes a harmonious environment when individual liberty is pursued thoroughly, as opposed to a more collectivist endeavor. This way, quality work can be produced in a timely manner.