Bash Script – What Does set -u Do?

A newline is ignored in a few contexts where there is manifestly an unterminated command. These contexts include after a control operator (&&, ||, |, &, ;, ;;, but not !).

I don't see this documented in the bash manual.

In POSIX, it's specified via the grammar rules. Wherever the rules have linebreak, you can have zero or more line breaks.

The documentation you quote gives three cases:

If bash is invoked with a file of commands, $0 is set to the name of that file.

(case 1)

If bash is started with the -c option, then $0 is set to the first argument after the string to be executed, if one is present.

(case 2; note the two "if"s, which must both be satisfied in this case)

Otherwise, it is set to the filename used to invoke bash, as given by argument zero.

(case 3).

The "otherwise" clause covers any situation which isn't covered by cases 1 and 2: bash isn't invoked with a file of commands, and bash isn't started with the -c option, or it's started with the -c option but without any argument after the string to be executed.

So yes, it includes the case where Bash is started with -c without any argument after the string to be executed. It also includes the basic echo $0 case when run from an interactive shell, since the interactive shell was most likely started without either a file of commands or a -c option.