Is the uncertainty principle a property of elementary particles or a result of our measurement tools?

The first paragraph is basically right, but I wouldn't ascribe the uncertainty principle to particles, just to the universe/physics in general. You can no more get arbitrarily good, simultaneous measurements of position and momentum (of anything) than you can construct a function with an arbitrarily narrow peak whose Fourier transform is also arbitrarily narrowly peaked. Physics tells us position and momentum are related via the Fourier transform, mathematics places hard limits on them based on this relation.

The second paragraph is used to explain the uncertainty principle all too often, and it is at best misleading, and really more wrong than anything else. To reiterate, uncertainty follows from the mathematical definitions of position and momentum, without consideration for what measurements you might be making. In fact, Bell's theorem tells us that under the hypothesis of locality (things are influenced only by their immediate surroundings, generally presumed to be true throughout physics), you cannot explain quantum mechanics by saying particles have "hidden" properties that can't be measured directly.

This takes some getting used to, but quantum mechanics really is a theory of probability distributions for variables, and as such is richer than classical theories where all quantities have definite, fixed, underlying values, observable or not. See also the Kochen-Specker theorem.

This is really a footnote to Chris' answer but it got a bit long for a comment.

It sounds odd to claim that a particle has no position, but it's easier to understand if you appreciate that a particle is just an excitation in a quantum field. When Heisenberg was developing his ideas physicists still thought of particles as little billiard balls. With the development of quantum field theory we now understand that a particle is just an excitation in a quantum field. For example there is an electron quantum field that pervades all of spacetime. If you add a quantum of energy to this field it appears as an electron. Add a second quantum of energy and you have two electrons, and so on. Similarly, take a quantum of energy out of the field and an electron disappears. Incidentally this also explains how matter can turn into energy and vice versa.

This means the objects we call particles are altogether stranger than Heisenberg thought. They are certainly not tiny billiard balls, and they don't have the properties associated with tiny billiard balls like a precise position and momentum. However, your second paragraph verges on the truth when it points out that when the electron field exchanges energy with something else the exchange takes place at a (reasonably) well defined point, and we can think of this as the position of the electron.

Although the uncertainty principle stems from the mathematical structure of QM, i.e., originates from the noncommutivity of some observable letting them behave as fourier transform pair as explained in another answer, I still think it is a statement on measurements, (i.e., imposes fundamental limits on measurements) since QM itself seems to be a theory of measurements (i.e., not of ontological reality).

I want to quote here the viewpoint of Asher Peres from his precious book, Quantum Theory: Concepts and Methods:

$$ \Delta x\,\Delta p\ge \hbar/2\,.\tag{4.54}$$

An uncertainty relation such as (4.54) is not a statement about the accuracy of our measuring instruments. On the contrary, its derivation assumes the existence of perfect instruments (the experimental errors due to common laboratory hardware are usually much larger than these quantum uncertainties). The only correct interpretation of (4.54) is the following: If the same preparation procedure is repeated many times, and is followed either by a measurement of $x$, or by a measurement of $p$, the various results obtained for $x$ and for $p$ have standard deviations, $\Delta\,x$ and $\Delta\,p$, whose product cannot be less than $\hbar/2$. There never is any question here that a measurement of $x$ “disturbs” the value of $p$ and vice-versa, as sometimes claimed. These measurements are indeed incompatible, but they are performed on different particles (all of which were identically prepared) and therefore these measurements cannot disturb each other in any way. The uncertainty relation (4.54), or more generally (4.40), only reflects the intrinsic randomness of the outcomes of quantum tests.