According to Apostol's ANT,
The smallest positive integer $f$ such that $$a \equiv 1 \mod m$$ is called the exponent of $a$ modulo $m$, and is denoted by writing $$f = \exp_m(a).$$ If $\exp_m(a) = \phi(m)$ then $a$ is called a primitive root mod $m$.
Then later the book says that
... primitive roots exist for the modulus 4.
For $m=4$ we have $\phi(4)=2$. If we suppose that $\gcd(a,m)=1$ then $a$ is any odd number. So we must show that $a^2 \equiv 1 \ \text{(mod m)}$ is possible and $a^1 \equiv 1 \ \text{(mod m)}$ is not. However for infinity many odd numbers of the form $a=4k-1 : a \equiv 1 \text{(mod m)}$ which means $f=1<2=\phi(4)=2$.
Where am I wrong? or is the text's mistake?
By your definition, $3$ is a primitive root$\mod 4$, because $3 \not \equiv 1 \pmod 4,3^2 \equiv 1 \pmod 4$, and $\phi(4)=2$. As you say, any integer of the form $4k+3$ would also qualify. The later statement that you quote, "primitive roots exist only for the modulus 4" is not correct. $2 \text { and }3$ are primitive roots modulo $5$. For example $2^2 \equiv 4, 2^3 \equiv 3, 2^4 \equiv 1 \pmod 5$ and $\phi (5)=4$. There are primitive roots modulo any prime and some composites. Please look around where you found that statement and see if there is some qualification.