Enamel erosion due to acid is tied to the acidity of the mouth, which, if the pH drops low enough, starts the demineralization of teeth (note, however, that this is a dynamic process, with systems in place to remineralize teeth, and that this dance happens every time we eat or drink something).
Typically carbonated water alone will be well under a pH of 4, which is itself well below the "5.0–5.7...known to trigger dental erosion effects" in the actually cited fact in that article.
"Dissolution levels with all of the mineral waters were very low and for several still waters were undetectable. Sparkling mineral waters showed slightly greater dissolution than still waters, but levels remained low and were of the order of one hundred times less than the comparator soft drinks. De-gassing of a sparkling mineral water reduced its dissolution, but the total levels were still relatively low suggesting that carbonation of drinks may not be an important factor per se in respect of erosive potential."
"The complex and heterogeneous mineral compositions of mineral waters could influence the dissolution equilibrium of apatite in enamel and controlled addition of several ions to ultrapure deionized water was investigated. Calcium ions led to the greatest reduction in hydroxyapatite dissolution, but their effects were moderated by other ions including magnesium and sulphate. Thus, mineral waters appear to offer a safe alternative to more erosive acidic beverages and their complex mineral ion compositions may positively influence any dissolution processes at the tooth surface."
This effect can be surprisingly strong, e.g. here's a study from a few years later showing "that fortifying apple, orange and grapefruit juices with calcium prevented enamel erosion and decreased root surface erosion (P < .01)."[1]
In fact buffering is well known to protect calcium phosphate in teeth from dissolution (e.g. an older study here[2] which points to the saturation of the same calcium and phosphate ions that the study you cite does as the source of protection), so it's actually quite odd that they would suggest that minimal dissolution from sparkling mineral water would suggest that the carbonation may not be an important factor of erosive potential.
It's hard to judge too much from just an abstract, however, and I can't find a full copy anywhere. That suggestion may just be an off-hand one.
Neutral pH is 7, things with a lower pH are more acidic Things with a higher pH are basic. So Carbonated water, with it pH of ~4 is a much stronger acid than something with a pH of 5
Carbonated water's acidity puts it at a lower pH than the acidity needed to start demineralization, so carbonic acid can cause enamel erosion.
Coke changes ingredients around to fit local regulations. In Europe, Diet Coke uses acesulfame potassium ("ace-K"), aspartame and sodium cyclamate. Cyclamate's effect is to neutralize some of the bitterness of ace-K and aspartame, but it's banned in the US and some other countries in the 1950s due to research determining it to be harmful. The ban is being upheld despite later research showing that the original science was flawed.
(Downvotes, really?)
But, in addition to phosphoric acid, Coke also has citric acid. I'm not sure you could remove both and get a drink that's still tasty enough...
EDIT: See below. Carbonic acid may not be as meek as I thought.