By "hard water", I mean a water that has a high concentration of minerals in it, particularly magnesium and calcium. Generally water high in dissolved minerals (Total Dissolved Solids) is "hard". A water with few dissolved minerals is "soft".
The terms "hard water" and "soft water" go back over 300 years, but I am not entirely sure why they are called by those names.
(There is an exception, which is water high in sodium or potassium. This water will "act" like a soft water in certain circumstances (like in making soap lather), but is high in dissolved minerals like a hard water. There are a few places that have water like this naturally, but generally this type of water is called "artificially softened" water. I'm not sure how it compares in tea making to true soft water. )
Okay, I did some more reading on hard and soft waters and have a better understanding of the accepted use of these terms. As a definition, the water I use is almost hard, TDS=242mg/L. The definition of a mineral water is at least TDS=250mg/L. This water, in general, makes better tasting tea than the purified Brita water that I used to use, in my opinion. This may be subjective. It also has a higher ph, 7.2 compared with my Brita filtered water at 6. The problem of my water is the scale which you would think is caused by the higher TDS.
The other mineral water that I use has a higher TDS, 358mg/L and slightly higher ph of 7.3. However, this water does not make my teas taste as good but has LESS scale than the lower TDS water I generally use. It also has half of the calcium contained in my regular mineral water.
The scale I'm willing to live with and clean out every so often. But, trying to get the best taste out of my teas is another matter. Perhaps it's the higher calcium in the TDS that makes greener teas sing. Maybe it's the higher ph. I don't know.
When you say softer water makes higher quality teas shine, this is not my experience. But, maybe there are some teas that do respond to softer water but I don't think it's the quality of the tea but the nature of a particular type of tea like Wuyi and its processing. I'm guessing here. I know when I switched over to a mineral water with a higher TDS, my tea experience heightened dramatically.
Stentor: Don't know the answer to your question about how high a TDS is too high. I only have experience with the 2 waters I mention above, the higher TDS one with a measurement of 358mg/L.