All mentions, that I am aware of, regarding Bladesingers are below:

• AD&D 2^nd Edition / Complete Book of Elves

  Distinctive Appearance. Not only are they easily identifiable by their weapon of choice and catlike grace, Bladesingers are decorated with their weapon guild's distinctive tattoo. Each guild has a separate and unique tattoo depicting its style of weapon through an animal representation. Long swords are often represented by great cats such as lions or panthers, a whip would be shown as a striking snake, and so forth.

• AD&D 2^nd Edition / Elves of Evermeet

  On Evermeet they are organized into small warrior lodges, each specializing in a different weapon. They often tattoo themselves with old runes, and each lodge has a special symbol it's members tattoo themselves with to identify their membership.

Bladesingers are mentioned in the edition after AD&D 2^nd Edition, and despite its contradiction to 'guilds' or 'schools' there is no mention of any particular weapon (besides blades) to be in use.

• D&D 3^rd Edition / Races of Faerun

  Most bladesingers work alone, sufficient unto themselves, but in larger communities they sometimes have the opportunity to fight together in the same combat. Bladesingers are normally trained singly by another bladesinger, and the concept of anything as formalized as a bladesinger school is an absurd notion to them.

What that means, in that edition, is a school representing animals went to the way side. in the revised edition, D&D 3.5 Edition, Bladesingers were mentioned as well.

• D&D 3.5 Edition / Complete Warrior

  Bladesingers are elves who have blended art, swordplay, and arcane magic into a harmonious whole.

It is essentially a copy paste, with a lot omitted, from Races of Faerun. Only "swordplay" and "any martial weapon" is mentioned. Nothing about schools guilds at all. There is another book the Bladesinger is mentioned, but its essentially only a blurb.

• D&D 3.5 Edition / Races of the Wild

  Bladesinger (Complete Warrior): Practitioner of the quintessential elven martial art, the bladesinger is both a skilled mage and a deadly swordmaster.

Once again, only sword is mentioned; probably due to racial sword proficiency - and the implication of blade in the name Bladesinger. Eventually, near the end of this edition, Bladesingers became Duskblades, and lost its 'elf-only' motif.

• D&D 3.5 Edition / Player's Handbook II

  While the ability to cast arcane spells in armor originated with the elves, over the millennia the secrets of the duskblade have been disseminated to the other races, and today members of any race can become a duskblade.

No mention of guilds or schools, or even preferred weapons were mentioned. Any martial weapon it appeared to be - but spending a feat to use a whip...most people would say, "no thank you." Bladesingers were included in the edition after this one, much to my surprise. But, it discredits the 'different weapons' aspect entirely.

• D&D 4^th Edition / Neverwinter Campaign Guide

  Benefit: Choose a one-handed melee weapon with which you have proficiency, and that is a light blade or a heavy blade.

Now we come to D&D 5^th Edition, and you already have the published information on them thus far. Since, to my knowledge, that edition is to harken back to the days of simpler game-play and more intrinsic role-play; rather than the vice versa, take this opportunity to "Create Your Own" schools, guilds, and organizations.

In later D&D editions, the term eldritch is associated with the warlock class, whose signature class feature is eldritch blast. The warlock was introduced in 3.5e’s Complete Arcane, and was a core class in both 4e and 5e. All three of these warlocks tended to revolve around the use of eldritch blast, and as a result, feats and items specifically tailored for this class often use the word eldritch, and such things that are more generally for all arcanists tend to avoid it.

The warlock class is an arcane class, so eldritch can be seen as a subset of arcane. However, warlocks generally gain their powers not through study (as with wizards) or ancestry (as with sorcerers), but through pacts with fey, fiendish, or alien creatures. This accounts for the connotations and associations you’re seeing around the eldritch term.

However, the term is also simply an English word, and it is sometimes used in situations divorced from the warlock class even after it was printed (and clearly, prior to Complete Arcane, any use of the term had nothing to do with the class that hadn’t been written yet). This is seen perhaps most notably (to modern D&D, anyway) in the eldritch knight class; that was originally printed as a prestige class in the Dungeon Master’s Guide for 3.5, before Complete Arcane was printed. At that point, it was just used as more-or-less a synonym for arcane. And even though both 4e and 5e already had warlocks with associations with the word eldritch, Wizards chose to keep the eldritch knight name for an arcane-casting fighter, using it as a knight paragon path in 4e and a fighter archetype in 5e. So the eldritch knight could be seen as a big exception to the idea of “eldritch” being associated with warlocks. There are others.

However, even when used simply as English words, the words have different connotations. Arcane means complex and/or secret, and anything secret can have sinister associations, but eldritch highlights them. Something arcane may merely be complex and difficult to understand (quantum mechanics is very arcane), but something eldritch more strongly hints at something sinister going on.

For reference, Google provides the following definitions:

el·dritch

ˈeldriCH

adjective

weird and sinister or ghostly.

"an eldritch screech"

ar·cane

ärˈkān

adjective

understood by few; mysterious or secret.

"modern math and its arcane notation"