Are these two quantum systems distinguishable?

These systems are not distiguishable. The average density matrix is the same, and the probability distribution obtained by performing any measurement depends only on the average density matrix.

For the first system, the density matrix is $$\frac{1}{2} \left[\left(\begin{array}{cc}1&0\cr 0&0\end{array}\right)+ \left(\begin{array}{cc}0&0\cr 0&1\end{array}\right)\right].$$

For the second system, the density matrix is $$\frac{1}{2\pi} \int_\theta \frac{1}{2}\left(\begin{array}{cc}1&e^{-i\theta}\cr e^{i \theta}&1\end{array}\right) d \theta.$$

It is easily checked that these are the same.

Case 1: $\frac{1}{2}\left[\left|0\right>\left<0\right|+\left|1\right>\left<1\right|\right]$.

Case 2, average over phases $0$ to $2\pi$: $$\frac{\int\left[(\left|0\right>+e^{i\theta}\left|1\right>) (\left<0\right|+e^{-i\theta}\left<1\right|)\right]d\theta} {\int\left[(\left<0\right|+e^{-i\theta}\left<1\right|) (\left|0\right>+e^{i\theta}\left|1\right>)\right]d\theta}.$$ The cross terms average to zero because $\int\limits_0^{2\pi} e^{i\theta}d\theta=0$, so it's the same density matrix. If this is really what the different manufacturers deliver, it's not a cheap knock-off.

Let me give some reference that might be useful to make things clear.
It's Landau-Lifshitz, book 5, chapter 5:

The averaging by means of the statisitcal matrix ... has a twofold nature. It comprises both the averaging due to the probalistic nature of the quantum description (even when as complete as possible) and the statistical averaging necessiated by the incompleteness of our information concerning the object considered.... It must be borne in mind, however, that these constituents cannot be separated; the whole averaging procedure is carried out as a single operation, and cannot be represented as the result of succesive averagings, one purely quantum-mechanical and the other purely statistical.

This "twofold averaging" is exactly the reason why the two states cannot be distinguished in any way.

Let me add another nice citation:

It must be emphasised that the averaging over various $\psi$ states, which we have used in order to illustrate the transition from a complete to an incomplete quantum-mechanical description has only a very formal significance. In particular, it would be quite incorrect to suppose that the description by means of the density matrix signifies that the subsystem can be in various $\psi$ states with various probabilities and that the average is over these probabilities. Such a treatment would be in conflict with the basic pronciples of quantum mechanics.