What is a 'nonexaggerated' word for egotistical?

(I've slightly reworded your first sentence.)

His behavior was recognized as egotistical, but in fact it was just healthy self-confidence.

We all need to be more self-confident and less egotistical or oblivious to our own wellbeing.

His self-confident behaviour was exactly what he needed to learn for his own well being.

self-confidence (n.)

Confidence in oneself and in one's powers and abilities m-w

self-confident (adj.)

Having confidence in one's abilities, qualities, and judgement; characterized by self-confidence. OED

Studies show that a high level of self-confidence is related to high levels of overall well-being and happiness. It has also been connected to creativity, which is a quality that can enrich relationships in numerous ways. S. Degges-White and C. Borzumato-Gainey; Friends Forever (2011)

You should be cautioned, however, to learn the difference between Self-confidence, which is based upon sound knowledge of what you know and what you can do, and egotism, which is only based upon what you wish you knew or could do. ...

If you have Self-confidence those around you will discover this fact. Let them make the discovery. They will feel proud of their alterness in having made the discovery, and you will be free from the suspicion of egotism. Napoleon Hill; The Law of Success (1928)

Self-Confidence versus Egotism

We have all met the man who "knows it all." How offensive he is whenever and wherever met...But he is eternally an unmitigated and unanimous nuisance.

It might be supposed that the medical profession, from liberal education should be free and exempt from these social and business plagues. But it is not so. ...

...we are glad to claim for our people what many regard as a synonym, i.e., self-confidence. Tho seldom recognized, there is a vast difference between egotism and self-confidence. we admit their kinship, but deny their identity...

The egotist has done nothing, while the man of self-confidence is the life and hope of the profession. The Medical World, Vol. 18, p.81 (1900)

If egotism be synonymous with self-confidence, every successful man is an egotist, for he dares to attempt what hundreds have failed to accomplish, and ventures to compete with real ability and skill. Rev. A. T. Pierson; "Egotism versus Self-Knowledge", Our Monthly, Vol. 4, p.220 (1871)