Physics uses math to describe the real world. That means that one has to distinguish between mathematically rigorous statements and what they mean in application to the real world, i.e., what they mean in experiment. This is particularly true for quantum mechanics and relativity, which, in essence, emerged from posing a question aboyt how we actually measure things and what can or cannot be measured. Notably, in quantum mechanics one can calculate the uncertainty of two non-commuting quantities for the same particle, but one can never measure them for the same particle.
From mathematical point of view, the uncertainty principle is a rigorous mathematical statement - essentially a restatement of Schartz inequality in the context of quantum mechanics. From physical point of view it is a statement about the asymptotic relation between the sample variances that are obtained after performing a very large number of measurements. (See for a bit more details my answer to a similar question.)
Thus, the most important thing here is realizing the distinction between the math and the nature - both types of the textbooks cited in the OP seem to take rather one-sided view of the things.