What is being stated about DASH is that its methods of "addressing fungibility improvements directly" are unsound. So unsound in fact as to deserve the term "snake oil."
Nowhere in your links for the definition of money is it stated that a requirement for being considered base money is that it have a public accounting system. Gold does not have a universal directory of past transactions. Nor does cash. You have not explained why it is essential or even preferential for a currency to have this property, and none of your links claim it to be necessary in their definitions.
Contrary to what Monero proponents like to promote, bitcoin IS a fully anonymous currency. Any information about who happened to be on the other end of a transaction has to be gleaned from OUTSIDE the blockchain. In this sense, it's exactly the same as paper cash - if you hand me a 100 dollar note, I can clearly associate that 100 dollar note with you.
What if I mail it?
This thread is about dash, but since you mention monero, there is a function called 'viewkey' which performs the exact function you claim it is not capable of.
The amount of information you can glean about the people controlling the movements is mitigated by maximising the fungibility of the base-monetary media (which, as I've explained, is exactly what Dash is doing).
Dash's methods of "maximising the fungibility of the base-monetary media" is what is being claimed to be unsound.
On the other hand, obfuscation of the blockchain is attaching a priority to the base monetary media that does not belong there - and particularly not in a new digital medium that portends to function as base money.
By your own definitions, obfuscation of the blockchain actually IS a requirement to be considered money. Fungibility (aka consistency) happens to be a requirement. Dash's unsound attempts to be fungible happen on a volountary basis. Unmixed coins used to buy CP are not the same as coins that have been mixed. By their taint they carry properties the mixed coins do not. The same goes for unmixed coins used for non-nefarious purposes. The point is that they are not consistent.
The 'public blockchain' is therefore an essential element of anything that portends to act as 'base money' as opposed to a record keeping system which is backed by such.
This is what you have not explained....