A byte is the smallest thing that a pointer can point at. Something like MIPS (from memory) has 8-bit bytes, but a load instruction loads 4-bytes at a time (and although it has the unaligned load functions which can be convinced to load a single byte, they aren't general purpose)
Before that you had CDC machines with 6-bit bytes (or Nybbles if you owned an Apple ][ with a floppy disk)
If I had to pick a date, I'd pick 1977 as when it solidified. A few elements: ARPAnet standardized on octet-based protocols in the early 1970s, and was getting widespread by the mid 1970s; three popular microcomputers based on 8-bit architectures were introduced in 1977 (the Apple II, TRS-80, and Commodore PET); and DEC introduced the 32-bit VAX in 1977.
But sure enough, I was very wrong on (A): Apple II disk writes were done in "sets of 5-bit or, later, 6-bit nibbles" [1].
Thank you, chiph!
[0] http://www.nibblemagazine.com/ [1] https://en.wikipedia.org/wiki/Nibble#History
"A byte is a set of 8 bits. Computers typically move data around a byte at a time."
A byte being 8 bits is ok. Historically, a byte might have a different number of bits, but all modern architectures use 8 bits. Since this is an introductory article, this is fine. (more details: http://en.wikipedia.org/wiki/Byte)
Computers do not typically move data in byte chunks. You could say "a byte is the smallest unit of data a CPU can load or store". If you talk about moving data, the question is between what. Probably memory. However, there are caches nowadays, since bandwidth is cheap and latency is expensive. Data is moved in cache line chunks, which means 4-64 byte chunks depending on architecture and cache level. Bigger chunks in upcoming architectures.