In Processing, using the default I/O functions, you have no choice: it will be saved in UTF-8 encoding.

" not so heavy"
Well, text files are rarely heavy, except when you write gigabyte log files or similar.
You can make text files heavier by choosing UTF-16 encoding, of course: each character will take 2 bytes!
UTF-8 is similar to Ascii in weight, ie. if you write only plain Ascii characters (no accents, no special chars), they are identical.
Ansi, if you mean ISO-5589-1 (or -15), is slightly more compact, but you will see a significant difference only if you have lot of high-Ascii characters in the files. For normal text, there is no difference, and the inconveniences are: no support for other code pages (eg. an occasional Turkish or Cyrillic character) and no way to know which code page is used (idem for UTF-8, but at least it is often identifiable algorithmically and visually).