I found the solution on this thread. It turned out that since 2011, vimrc_example.vim was updated to include this textwidth=78 rule as an autocommand, and my _vimrc sources vimrc_example.vim (probably because I based it on the original — which means this situation will apply for many users).
One solution is to find the line in your .vimrc, that says
source $VIMRUNTIME/vimrc_example.vim And after that line, insert the following line to unset the particular autocommand that's making a nuisance:
au! vimrcEx FileType text How I found the solution: I suspected that it was an autocommand, so I tried looking through all autocommands by using the :autocmd command. But there were just too many. To simplify things, I inserted that output into a spare buffer, as described here. Searching for tw=78 showed the culprit autocommand with the keyword vimrcEx, and then asking Google for the keyword (or I could have grepped Vim's directory) lead us to the solution.