Hypothesis 1 is a comparison of a sample mean (the mean of $X_1-X_2$) to a theoric value of $0$:
H0: μd = 0 H1: μd ≠ 0
The test variable will therefore be $\frac{m_d}{\sqrt{\frac{\sigma_d^2}{n_1}}}$
Hypothesis 2 is a comparison of 2 sample means (the mean of $X_1$ and the mean of $X_2$):
H0: μ1 = μ2 H1: μ1 ≠ μ2
The test variable will therefore be $\frac{m_1-m_2}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}$
Please note that the 2 hypothesis can only be compared if $n_1=n_2$, else the mean of $X_1-X_2$ have no sense.
In the numerator, we have the same thing since $Mean(A-B) = Mean(A)-Mean(B)$.
But in the denominator though, things may be different because $var(A-B) = Var(A)+Var(B)+2\times Covar(A, B)$.
Therefore, tests will be the same only if $A$ and $B$ are independant, so $Covar(A, B)=0$.
I hope this helps you to understand why (and moslty when) these hypothesis are different.
Maybe thinking in terms of distributions instead of means will help too: the first hypothesis is testing if the distribution of a variable (which is the difference between 2 variables) is centered on a value significantly different from $0$, while the second is testing if there is a significant difference between the 2 distributions of 2 variables.
As @Henry said, the hypothesis #1 is especially adapted to the paired scheme, where $n_1=n_2$ by concept and the very variable of interest if the difference in the pair more than the difference between the groups.
