Is the statement "$p$ implies $q$" logically equivalent to the statement "$p$ implies only $q$"?

Question

I am confused if the statement "$p$ implies $q$" logically equivalent to the statement "$p$ implies only $q$"?

Assuming that the two said statement is logically equivalent, then the truth value of the statement ...

"If $a^2=b$ and $b>0$, then $a=\sqrt{b}$."

... is false. Since a can be equal to $a=\sqrt{b}$ OR $a=-\sqrt{b}$, not only $a=\sqrt{b}$.

Answer 1

$$a^2=b>0\implies a=\sqrt b$$ is a false statement, full stop.

$$a^2=b>0\implies a=\sqrt b\lor a=-\sqrt b$$

is a true statement, full stop.

You never assume that "there could be other predicates but they are missing" or anything of this kind. If you want to express that "$a=\text{only}\sqrt b$" with the ulterior motive that "$a=-\sqrt b$" could have been possible, you write $a=\sqrt b$, and there is no need to mention $a\ne-\sqrt b$.

Answer 2

They are not equivalent. For example the statement "If it's raining, then I'll use an umbrella" makes sense but "If it's raining, then only I'll use an umbrella" is false since others may use an umbrella as well.