The research article mostly talks about isotopic ratios for some nobles gasses, and they discuss the presence of certain xenon isotopes in excessive amounts, which is an indicator of the age of the sample. Indeed, they're characteristic of an atmosphere of at least 1.5 billion years ago. These elements are tasteless, though, and won't make a noticeable difference.
In the link, the geologist said it's salty. Why is that? Saltiness is one of the basic tastes, along with sweetness, sourness, bitterness, and umami (also known as 'savoriness'; which responds to nucleotides and glutamate, commonly found in cooked meat, fermented products, and MSG-flavored food). In our mouth, there are many receptors that are activated by alkali metals. The simplest and most known is the one for sodium chloride (common table salt).
Sodium alone is mainly responsible for activating this receptor. How? Sodium can enter the cell, thus depolarizing it (reducing difference in electric charge) and this is a signal for some voltage-dependent calcium channels, which in turn trigger the release of neurotransmitters making you perceive the saltiness. So, sodium enters -> changes voltage in the cell -> triggers release of neurotransmitter -> your brain considers that 'salty'.
But, as I said, other ions can also do this, with the caveat that—as a rule of thumb— the further away they are in the periodic table from sodium, the less salty they taste (this is why KCl, a common salt substitute for people with hypertension, doesn't taste quite as salty).
So why is that water really salty? Given what we've learnt, that can be due to a combination of two things:
- many metal cations in the water
- sodium is abundant