All chlorine products add something to the water in addition to the chlorine itself.
Trichlor adds 6 ppm CYA for every 10 ppm of chlorine.
Dichlor adds 9 ppm CYA for every 10 ppm of chlorine.
Naturally, both of these will elevate CYA pretty quickly, especially dichlor.
Calcium Hypochlorite is about 1/3 calcium, so it will raise the calcium levels. It should not affect the CYA levels. As to the 27% of 'other' ingredients in the Leslies product, all I know is that it is inert, and for most purposes can be ignored.
Sodium Hypochlorite will add some salt to water in addition to chlorine, but the levels are pretty small and has never been found, to my knowledge, to be a measurable problem in any pool, as virtually any splashout or rain will counter it.
Lithium Hypochlorite may add some metals, notably lithium, but it's so expensive that I don't know anyone that uses it. Or could afford to use it for long. It may be the perfect sanitizer, or it may be cheaper to treat the water with weapons-grade plutonium, for all I know. You're on your own with this one.
So... why all the different products?
The tabs are really convenient. They can be placed in an erosion feeder and left to do their thing. They are are easy to handle, easily stored, and have a good shelf life.
Powders are easily broadcast, and pack a lot of "chlorine power!!", are easily stored and have a good shelf life.
Liquids are more cumbersome to store and handle, and are more sensitive to loss of efficacy from sun/heat/time.
As you can see, there are trade-offs with everything. For us, sodium hypochlorite (liquid chlorine, or bleach) is the simplest and best product with the least side effects of all of the available chlorine sanitizing products. Can the others be used? Certainly, if one knows about and is prepared to deal with the additional effects, such as drain/refill for trichlor/dichlor/cal-hypo, and the ultimate bank account vacuuming that comes with lithium.
uhwelliuh - not sure what to tell you about your CYA levels and the calhypo. My first instinct is to suspect the testing. CYA tests are, at best, plus/minus 10, and testing variations of time, light, reagent age, procedure consistency, as well as the person doing the testing can all impact the result. Variations of 50 or more between the same test performed at different pool stores are common.