In many physics divulgation books I've read, this seems to be a commonly accepted point of view (I'm making this quote up, as I don't remember the exact words, but this should give you an idea):


  Heisenberg's uncertainty principle is not a result of our lack of proper measurement tools. The fact that we can't precisely know both the position and momentum of an elementary particle is, indeed, a property of the particle itself. It is an intrinsic property of the Universe we live in.


Then this video came out: Heisenberg's Microscope - Sixty Symbols (skip to 2:38, if you're already familiar with the uncertainty principle).

So, correct me if I'm wrong, what we may claim according to the video is:


  the only way to measure an elementary particle is to make it interact with another elementary particle: it is therefore incorrect to say that an elementary particle doesn't have a well defined momentum/position before we make our measurement. We cannot access this data (momentum/position) without changing it, therefore it is correct to say that our ignorance about this data is not an intrinsic property of the Universe (but, rather, an important limit of how we can measure it).


Please tell me how can both of the highlighted paragraphs be true or how they should be corrected.

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.
