Does concentration or pKa define acid strength?
The term ‘strong acid’ is sometimes used in a rather fuzzy way and you ran into problems doing so. I prefer to use the term ‘strong acid’ only with respect to an acid’s $\mathrm pK_\mathrm{a}$ value and disregard all other influences. This gives a clearly defined measure of acid strength and we can easily sort various acids by their strength into stronger or weaker acids.

However, this is looking at the acid as a molecule. In real-life applications you are typically more interested in the property of a solution. To give a real-world example, imagine a $\pu{1M}\ \ce{HBr}$ solution and a $\pu{12M}\ \ce{HCl}$. solution. Obviously, $\ce{HBr}$ is the stronger acid, but the concentration of $\ce{HCl}$—also a strong acid and thus fully deprotonated—is higher. Therefore, the $\ce{HCl}$ solution is more concentrated or, as some would say, more acidic. It can do greater harm and it is able to protonate more Brønsted base molecules than its $\ce{HBr}$ counterpart.

If instead of examining a $\pu{1M}$ and a $\pu{12M}$ solution I had examined a $\pu{e-3M}$ and a $\pu{1M}$ solution, we could even base the discussion around the solution’s resulting pH: the stronger acid is much more diluted and will result in a solution of pH 3 while the weaker acid results in a solution of pH 0.