Modern medieval combat enthusiasts love pointing out that the public image of medieval swords is the opposite of the truth. That they were not sluggish, blade-heavy heavy weapons designed to be effective against armour but instead light and well balanced weapons for use against unarmoured opponents. While this is certainly true most of the time there were actually plenty of swords in the high middle ages that did very much match the former description! I'd say most of them are members of Oakeshott's type xiii family though there were also plenty in the type xii group, the most famous of which I can think of is the 13th century Saint Maurice sword in Turin. I take issue with these sword being grouped together with classic knightly swords. The arming sword I think is the light and nimble weapon that medieval enthusiasts (including myself) don't want this medieval sword stereotype put upon because it is quite the opposite. So I think it makes sense for a new classification to be made for these weapons that do fit the description currently embedded in the public mindset about what medieval swords were.

A commonly quoted quote is Oakeshott's "Medieval Swords are neither unwieldably heavy nor all alike - the average weight of any one of normal size is between 2.5 lb. and 3.5 lbs. Even the big hand-and-a-half 'war' swords rarely weigh more than 4.5 lbs. Such weights, to men who were trained to use the sword from the age of seven (and who had to be tough specimens to survive that age) , were by no means too great to be practical."

Compared to claims that Medieval European swords weighed 20lb, saying (accurately) that they tended to be 2.5-3.5lb is indeed saying they were light and well-balanced. But it is reasonable to be concerned about the pendulum swinging too far the other way. There are antique swords that, on what I hope would be a reasonable objective scale are not light, or aren't well-balanced, or perhaps even neither. For one-handed swords, this appears to be most common for cavalry swords, where the intent is not to engage in "fencing", but rather to hit the opponent as one rides past. Still nothing like 20lb monsters, but more likely to be 2.5-3lb relatively blade heavy swords. I once described a sword with weight and balance similar to the Turin St Maurice as an "indelicate pig of a sword". Plenty of 19th century European cavalry swords of which the same could be said.

I don't think that these are in the usual category of "arming sword". (Lacking a consistent and explicit definition of 'arming sword", it's hard to be certain.) They do appear to be "knightly swords". To speak of "arming swords", "cavalry swords" (or whatever might be a better name), and other specific types, as sub-types of "knightly swords" might be a good idea.

The thing is that you are missing that there were two distinct types of combat in the Middle Ages... on foot, and on horse. They have different dynamics and physics. And swords were made for (mostly) the upper classes... people who would be, more often than not, fighting from horse-back. As such, at least within the period you mention, swords were largely oriented towards such a fashion of combat. Fierce cutters, somewhat blade-heavy, well suited for large shearing attacks against lightly armoured opponents.

The Oakeshott typology works fine; it's just perhaps a lack of knowledge on your part. There are a few tweaks that could be made, but in general as the man himself said, it's a guideline rather than a hard-and-fast rule.

I certainly understand the frustration with the huge misconception of the mass and functionality of medieval swords. I am always assuring people who "...can't believe how light that sword is!" and my favorite: "Wow, that's so light! how much did a REAL sword weigh?"
There's nothing that changing sword classification can do that will change the public's misconceptions (ignorance) about swords. I usually tell people that the earlier swords of the Migration Period (actually, depending on the audience, I often say "dark ages") were much heavier and less balanced. But as technology allowed, steel improved and better steel meant that swords could be be thinner and lighter.
As for my thoughts on how heavy many of the xii's and xiii's were, I even find the type xiv's are too weighty for sword and buckler training. Of course, swords get heavier every year since I passed 60.
Today, I use a sub-two pound type xviii from customswordshoppe.com and a fist buckler (basically just a shield boss with a handle in back) for sword and buckler.
Just shows that not many knights or fencers made it to their 60's back in the middle ages.