Which bit goes first

First in goes furthest

Every clock edge pushes the bits one place along, from Q0 towards Q7. So the first bit you send is pushed eight times and ends in Q7, and the last bit sent stays in Q0. On this board Q7 is the dot and Q0 is segment a.

shiftOut(dataPin, clockPin, order, value) takes the order as its third argument:

• MSBFIRST sends bit 7 first. Bit 7 ends on Q7, the dot; bit 0 ends on Q0, segment a. That is the book's table.
• LSBFIRST sends bit 0 first. Bit 0 ends on the dot, and every other bit lands one place mirrored.

One word, every digit

Mixed up, the byte for 2, 0x5B, arrives as 0xDA: the dot lights and the shape is not a 2. Every digit comes out wrong in its own fixed way, and that consistency is the clue. Random segments that change from one attempt to the next are wiring; the same wrong shape every time is bit order.

A few bytes read the same both ways: 4 is 0x66, 0b01100110, a palindrome. So a sketch that shows 4 correctly can still have the order backwards. Check with 2.

The page this book replaces shipped a sketch with LSBFIRST and a table written backwards to match, segment a in bit 7. It worked, and it is the reason copying half of it into another sketch fails. The book's sketch uses the ordinary table and MSBFIRST.