We had those, too, but we never ever called them readers. It was always “The Reedly Weaker.” Always.
After a while, even the teachers called them that. 
We called them “readers” in the suburban Cleveland (OH) schools I attended from 1964 to '77. Some people might know them as primers (pronounced to rhyme with “trimmers” in my neck of the woods, although I’ve also heard the word spoken to rhyme with “timers”). I remember Ginn Basic Readers, but a better series was published by Holt, Reinhart and Winston. Titles included Sounds of Mystery, Sounds of the Storyteller, Sounds of a Distant Drum, etc.
Ah, SRA (Science Research Associates) cards! Yeah, it wasn’t too hard to figure out the order. When I was in fifth grade, for example, the lowest level was tan, while the highest was aqua. Kids know the relative academic merits of themselves and their classmates pretty well after a year or two of schooling.
I didn’t but that’s what our reading books had as subtitles on the covers when I was in elementary school. [insert more clever title in bigger, bolder text] First Grade Reader, [insert more clever title in bigger, bolder text] Third Grade Reader…
In the early years where I went to school, they were called primers.
Readers, from 1971-1980 in St. Louis. We had the pretentious single-word titles, too, though the only one I remember offhand is “Serendipity.” Generally they were dull as dust, the only thing duller was the occasional grammar chapter, which was doubly annoying since we also had English Books that were nothing but grammar and punctuation and such. Why did we do it twice? I have no idea.
Also annoying was the way that a certain percentage of “stories” were actually chapters of longer works. Sometimes really good longer works, like we read the Doldrums chapter of The Phantom Tollbooth. Could we read the whole book in class, instead of the other crap? Nope, just a single chapter to tease and torment you–“Yes, children, there are actually things out there that are fun to read–but that’s not what we’re going to do for class! Mwahaha!” And then when it wasn’t anything really good, it was a single chapter of something that was boring and then didn’t actually end or make any sense since the rest of the story wasn’t there.
We had SRA cards, too, but they weren’t booklets, just folding cards with very short stories. Most of them were dull as dust, too, although some of the stuff at the high end was pretty cool–I remember running across some Stephen Leacock when I was in 7th grade, and a few other things nearly as good. But the stuff you had to plow through to get there was awful.
My daughter, in first grade, doesn’t have a reader. They have short standalone books, some of them dull, some of them very good–she has to read one to me every night for homework.