Is this c the same as that c? Are the highlighted $c$'s the same or should it be $c_1$ and $c_2$.


I am not very good at analysis, but the answer to your question is clearly "the $c$'s are the same", by plain elementary logic. I'll try to explain it to you:
call $\varphi(\alpha)$ the "sentence"
$$\forall x \ \left[ \overline{\lim}_{r \to 0} \mu(B(x,r)/r^s < \alpha \right]$$
and call $\psi(\beta)$ the "sentence"
$$\mathcal H^s(F) \geq \mu(F)/\beta.$$
Then your theorem can be written as
$$\varphi(c) \implies \psi(c).$$
If you think about it, you can see that it would make no sense to write something like
$$\varphi(c_1) \implies \psi(c_2).$$

I'll use an easier example for you to understand better:
suppose $\alpha,\beta \in \mathbb N$ and call $\varphi(\alpha)$ the "sentence"
$$x \leq \alpha.$$
Now call $\psi(\beta)$ the "sentence"
$$x \leq 2\beta.$$
Then it is of course true that 
$$\varphi(n) \implies \psi(n),$$
since this can be written in natural language as

If $x \leq n$, then $x \leq 2n$.

But which sense would it make to say $\varphi(n_1) \implies \psi(n_2)$? That is, which sense would it make to state

If $x \leq n_1$, then $x \leq 2n_2$.

?
