The electron acts simultaneously from all the positions it occupies, and with a intensity and phase (or complex amplitude) given by the wavefunction.
Sometimes you consider interactions that can occur only in a volume smaller than the electron's delocalization (which is the atom's volume, for hydrogen). This is typically the case if the other interacting particle is more localized than the atom's electron, and the interaction is efficient in a small volume only. Then the atom's electron can interact only from a fraction of the volume it occupies, and the probability that you observe such an interaction drops as a consequence of the fraction of the wavefunction able to interact.
In that sense, the wavefunction (squared) tells a probability of interaction, what many people call a "probability of presence" - but please remember that the electron is simultaneously in all its possible positions. Quantum mechanics, as it is written and taught, carries a heavy load of historical misinterpretations that are presently abandoned.
The square of the wavefunction can be computed per volume element (for instance per pm3) directly. This is a function of x, y and z. For instance for the 1s, 2s... wavefunctions, it is biggest at the nucleus. Or you may consider a probability as a function of the radius to the nucleus, whatever the direction, as Hyperphysics does at your link. It's a density per length element, for instance per pm. For a spherical wavefunction, the density per radius unit is a product of the surface of the sphere of this radius, and then all such functions are zero at the nucleus.
"Both a wave and a particle": beware the historical misinterpretations there too. All interactions happen within a finite volume, they are never points. From a classical particle, quantum mechanics keeps very few attributes. For instance if an electron interacts by its charge, the interaction force or energy are computed for the whole charge at every possible position, rather than by spreading a charge density over all the occupied volume. Same for the momentum, kinetic energy... if they are defined well enough, an electron interacts with its full momentum and kinetic energy even if the interaction happens in a volume smaller than the electron's delocalization. Such attributes resemble better a classical particle if you wish.