Three separate things

Every command has three streams open: input, output, and errors. They look identical in a terminal because they both land on your screen, which is exactly why the distinction is easy to miss until it matters.

A pipe carries standard output only. Errors go straight past it to the terminal, which is why a broken pipeline still shows you the error.

Redirections are applied in the order they are written, and that order is the whole behaviour.

2>&1 >out first points errors at wherever output currently goes — the terminal — and only then moves output to the file. >out 2>&1 moves output first, so errors follow it into the file. They look like the same thing and are not.

This is a recurring defect in generated scripts, and it is pure order of operations.

A redirection on a command lasts for that command. exec applies one to the shell, and it lasts for the rest of the script: exec > run.log at the top means everything after it is captured without repeating >> run.log on every line.

It is also why a script that does it goes quiet. Nothing is broken and nothing failed — the output is in the file, and the terminal will not see another word of it.

exec cmd is the other half of it: the shell is replaced by the command, so nothing after that line runs. exec "$@" is the last line of most wrapper scripts for exactly that reason.