The parameters like GH, KH, pH and TDS are all interlinked. Afterall, we are talking about water. Those are just measurements of certain elements that are IN the water. They are all measurements of a certain parameter that make up the whole chemistry of water.
pH is a measure of how much Hydrogen (H) is in the water ("H"20) - the more H is in the water the more acidic it is. Conversely, the less H in the water the more alkaline it is.
GH - as mentioned, is a measure of Calcium and Magnesium (mostly), the test kit does get influenced by other minerals.
KH - as mentioned is a measure of Carbonates and Bicarbonates.
TDS - is a measure of Total dissolve solids. That is, a measure of everything else that is soluble in water. But it does not tell you what is in the water specifically. That's why we never rely on just one measurement. You can't rely on just TDS. Likewise you can't rely on just pH. It doesn't tell you the whole story.
So when you are talking about softwater, it means a low measurement of ALL those parameters. You cannot have a high pH but low KH. Or a high TDS and still expect a low GH. When you have a low TDS, it generally means you have low Calcium & magnesium (measured with GH kit), low Carbonates (measured with a KH kit). And because KH is low, pH will hence be low as well.
When people talk about one parameter like pH only, it either means that they emphasise it too much and don't understand the relationship with other parameters, or they don't have any other test kit apart from a pH meter/test kit. Either walk away, or you can try educating them. If you rely on just pH as an indicator of softwater, you might still fail to provide a softwater fish with the right environment. It might be right, as that is one parameter we measure to indicate "softwater". But what if you have a low pH, GH and KH but TDS is super high like in the 400's? It means that the water is full of another chemical or mineral or dissolved metal that your test kits are not picking up. If pH is only one part of the story, than knowing the other parameters means you have a much clearer picture of the story.
Those are only the 4 most common test we do in our hobby. That are many, many other tests that measures ONE specific parameter in water. You already know about the 4 mentioned above, we also commonly measure nitrogen ( ammonia, nitrite, nitrates), we have test kits for just Calcium, for just Magnesium, there are test kits for Oxygen, Iron, .... the list goes on.
So when we say "softwater", what do we mean specifically? That is the question.
For a tetra or dwarf cichlid that comes out of the Amazon river? Those are from softwater, yes, but Blackwater more specifically. The tannins and humic acids that make up the Amazonian blackwater is a big part of recreating the fish's environment. I guess what I'm trying to say here is that the term "softwater" is very generic. So if you need specifics, than you need to ask for specific parameters.