It's the same problem because the low scale matches in both definitions; and the high scale matches in both definitions, too. Both problems are the puzzle why the two scales are so much different.
First, the low scale. In the Higgs fine-tuning, you define the low scale as the Higgs mass. But the Higgs mass can't be parameterically greater than the Z-boson or W-boson mass. If it were much greater – assume the Standard Model – than the quartic coupling for the Higgs would have to be much greater than a number of order one and the perturbative series for the Higgs self-coupling would break down. In fact, at a slightly higher energy scale, due to running, one would run into the Landau pole and the coupling would diverge.
So as an order-of-magnitude estimate, the Z-boson mass, the W-boson mass, and the (lightest) Higgs boson (and vev) have to be of the same order and we call it the electroweak scale.
Now, the high scale. In your definition of the hierarchy problem, you just define it as the Planck scale. In the Higgs fine-tuning problem, you don't define the high scale explicitly but it's the scale of new particles or effects whose masses affect the Higgs mass via the quadratic divergence.
So whenever you have something like a particle of mass $\Lambda$, its loops connected to the Higgs in some way shift the squared Higgs mass by terms of order $\Lambda^2$. Clearly, the effects connected with the highest value of $\Lambda$ are the most important, dominant ones. The Planck scale, or slightly beneath the Planck scale, is the highest energy scale at which quantum field theory of some sort should hold. That's why it's legitimate to substitute the effects from this scale to $\Lambda^2$ and say that they contribute $m_{Pl}^2$ to the squared Higgs mass. Other effects contribute as well and the question is why the total Higgs mass is so much smaller – the squared Higgs mass is $10^{30}+$ times smaller than the squared Planck mass.
One can't believe any quantum field theory at energy scales exceeding the Planck scale because that's where gravity becomes strong and one needs a full theory of quantum gravity – probably synonymous with string/M-theory – which is strictly speaking not just a quantum field theory and the naive "addition of $\Lambda^2$" and similar QFT wisdom can't be relied upon.
