The thing to understand about LSI versus CSI is that LSI is a phenomenological model that was built by simply looking at how calcite formation (scale) was affected by various water parameters (pH, alkalinity, dissolved solids, temperature, etc). It's akin to a regression analysis or statistical model. CSI, on the other hand, is a model of calcite formation that is based entirely on "first principles" meaning that it is a quantity that is derived from the actual underlying chemical and physical properties of calcite formation. Both indices look at the same outcome ("does scale form or dissolve?"), they just take different approaches to get there.
TDS is related to amount of "stuff" dissolved in water. If you take a sample of filtered water (to remove suspended solids) and then gentle evaporate away all the water, what is left are the "dissolved solids". These dissolved chemical species affect or modify how water dissolves or forms calcium scale through their modification of the ionic strength of the solution. However, because field technicians and people in the industry are not trained chemists and don't really care about the atomic details of what's going on, or the physics of it for that matter, they desire to just have simple parameters to measure and then use a phenomenological approach to arrive at an answer that's "good enough" to direct their activities. The details don't really matter to them. Back in the old days, there were no easy field methods for measuring every individual chemical component found in water and, because water analysis often involve samples drawn from lakes & rivers, trying to figure out every particular dissolved chemical in the water would be a very difficult task. So TDS, which is often measured by proxy using electrical conductivity, was used to gauge how "pure" or "impure" the water sample was relative to the need (boiler fill water, cooling tower water, etc). If the TDS was too high for the intended application, then that would signal to the operator that the water needed to be treated first.
In the old pool management days, operators rarely ever measured things like cyanuric acid or calcium hardness. They only wanted to know how "contaminated" the water was so they could either dump it and start over or keep going and add chemicals to it. TDS was used as a proxy for that analysis. If the water in the pool had been there for several seasons with lots of sanitizer and other chemicals going in but not a lot of fresh water to dilute it, then the TDS would be high. That would signal the operator or owner to probably consider dumping the pool. Lots of lore and old-wive's tales built up around that in terms of TDS telling you that your sanitizer wouldn't work or that your pump would die, but none of that was ever grounded in reality. TDS is simply not that informative of a parameter now that people can measure all sorts of chemicals species independently.
Operators and industry types like Lowry that hype TDS simply don't understand the underlying physics and chemistry of what's going on. They do it because their target audiences are typically service techs and pool "boys" that need very simple concepts to do their weekly rounds. It's unfortunate that it spills over into the residential world of pool ownership and makes people think there is something special about it...As a variant of Clarke's 3rd Law states -
Any technology, no matter how primitive, is magic to those who don't understand it.
