This PDF from Southern Water in my view summarises fairly the difference between (i) salt-based softeners, (ii) phosphate-based ones, and (iii) "physical-softeners".
https://www.southernwater.co.uk/Media/Default/PDFs/2835b_WaterSofteners_A4Notes_Nov12.pdf
(By the way, I hold a different view on the sodium-health issue, a more conventional one in line with that of scientific-advisory councils around the World. As far as I can see, the evidence is strong that a high-sodium diet leads to higher blood pressure, which in-turn leads to increased cardiovascular disease, all on a population-wide basis (@JSHarris result not withstanding). Cardiovascular disease is a major cause of death. (Degree in physiology and PhD in biochemistry).
Interesting thread. I plan to add a salt-based water softener to my build, when I get to it (and a non-softened tap for in the kitchen for cooking/drinking).
Having done so past research on water softeners some time ago, I am tempted to use an older design of softener, a single-tank type, and one that has a tried-and-tested mechanical mechanism for timing and managing the resin ionisation step (no power supply needed). My research suggested to me that the modern two-tank systems are a marketing gimmick. Anyone can tolerate a few minutes of hard water once in a while if one attempts to draw water while the tank is regenerating. The added complication of two tank system provide only that marginal benefit for rather more complication. Having said this, it seems from what I have read that most salt-based water softeners are low maintenance (other than feeding them salt), no matter the type, so its probably marginal.