Something I wanted to point out is that even if you're just referring to the length of the sword's handle and not the blade, a longer handle still has the potential to get in the way or catch on your arm/other things more often than a shorter handle would. I've usually just assumed that whether a knight had a sword handle long enough for one hand, two hands, or just one and a half hands mainly just came down to personal preference.
I suspect the growing popularity of two-handed swords and other two-handed weapons during the late middle ages did have to do with the increasing frequency/effectiveness of metal armor. Although while i can't comment much on earlier centuries, by the late 1400s and 1500s two-handed or long-handled swords definitely don't seem to have been uniquely associated with knights or nobility. The swiss pikemen in particular were known for carrying longswords throughout much of the 16th century. Additionally, a lot of depictions of the new types of light/medium cavalry showing up in the 15th century seem to show them carrying fairly longhandled swords. For instance the ones in Daniel's blog post:
http://kriegsbuch.blogspot.com/2016/01/the-ap...n.html?m=1
In Feselen's Alesia painting some of the infantry are depicted fighting in the melee with two-handed swords but most of the fully-armored men at arms seem to be using short-handled swords:
https://commons.wikimedia.org/wiki/File:Siege_of_Alesia.jpg
(As an aside I keep noticing new things about the "old vs new" themes in the painting. Today it's the HRE infantry in the background assaulting the rounded bastion while
taking advantage of its blind spot)
Heavy cavalry carrying more than one melee weapon does seem to have been pretty common and it may have sometimes been both a one handed sword and a longsword. Montluc mentions that in the earlier parts of his career Gendarmes would each carry a broad, cutting falcion which was supposed to be able to cut into helmets and
mail armor. Although by the second half of the 16th century at least, english writers tend to specifically identify men-at-arms with carrying a mace at their saddle in addition to their sword, rather than any particular kind of sword.