Decide if a specific Turing machine halts on a specific string

Can you always decide if a specific Turing machine accepts a specific string?

If this is true, intuitively, it seams it would be implied that HALT could be decided, which is obviously not true. Otherwise what is the difference?

$$L_{HP-M'-x} = \begin{cases} 1 & \text{M' halts on x} \\ 0 & \text{otherwise} \end{cases}$$
The reason this language is decidable is because, given any $M'$ and $x$, this language must be either $\{1\}$ or $\{0\}$. In particular, this MUST be a finite language, and ALL finite languages are decidable.