The level of GH (hard water is defined as a high GH) depends only on those ions that precipitate with soap to make soap scum. In order of decreasing concentration in most tap water, this means: calcium, magnesium, iron (+3), manganese (+2). The concentrations of calcium and magnesium are much greater than the other two, so usually they are all we think about as far as the GH reading. A water softener has an ion exchange resin - the hard water ions attach to it, releasing K+ or Na+ to balance out the charge. When it is recharged, a brine (high KCl or NaCl conc solution) is flushed through it to remove the hard water ions and replace them with the K+ or Na+ for continued use.
Normal tap water will contain some potassium - again depending on the source, total dissolved solids, etc. It is impossible to say if hard water contains more potassium since hardness and potassium are totally separate things.
72g bowfront planted, CO2, 4x - T5HO, Eheim 2213 and 2217, 2 angels, pristella tetras, blue tetras, betta, albino bristlenose pleco, albino cories. Sword, vals, hygros, ludwigias, java moss and fern, anubias
2g Mac-quarium. Clown gravel, fluorescent plastic plants, and 2 guppies.