I'm looking for a book to serve as a "refresher" on  King Arthur and his knights---something that stays pretty close to the original tale, with the characters and relationships explained and/or re-told in contemporary language.

Nothing as "re-imagined", nothing re-interpreted as Marion Zimmer Bradley or Mercedes Lacky's new one, or most modern novelists.

 and not White's version. . I like them and I have plenty, but that's not what I'm seeking now.

I just want something relatively easy to read, and tells the stories as closely as possible to the original.

How about Sir Thomas Mallory's Le Morte d'Arthur? 

Or, The Once and Future King by TH White?

Mallory is great, but I was hoping for something a little more contemporary---like putting Mallory into modern language. And maybe briefer than White, although White would be very close.

There's a cute YA series that starts with The Squire's Tale by Gerald Morris you may enjoy. 

I would go with Rosemary Sutcliff's trilogy (YA level, and not too long)

The Sword and the Circle ~~ The Light Beyond the Forest ~~ The Road to Camlann.

How about Mary Stewart- The Hollow Hills and The Last Enchantment?

I love Mary Stewart's series, but I was looking for something with "less" of the writer's take on it.

Maybe I should look for Cliff's notes!

Rosemary Sutcliff.  Seriously.

I recently finished Sword at Sunset by Sutcliff - I thought it was excellent. This is one of the few, perhaps only, adult books she wrote.

Sutcliff presents a more realistic story than some Arthur legends, removing Merlin and many of the fantastic elements, and grounding Arthur (Artos) and his followers as clinging to Roman ways after Rome has left Britain to fend for itself. The battles in particular are described quite realistically.

Another excellent book by Sutcliff is Black Ships Before Troy.

The Pendragon by Catherine Christian

Look at the description for isbn 1566198763, the illustrated encyclopaedia (the title uses the old spelling)of Arthurian Legends by Ronan Coghlan.  Would this be helpful?