That is correct. The same goes in Hong Kong for their banknotes from different banks, Scotland for their different banks, Ireland for theirs, etc.
The UK has some different notes issued for places such as Jersey, Guernsey, Isle of Mann, Gibraltar, as well as Scotland and Ireland, that all have seperate notes. While these are pegged to the GBP, I've read in forum threads that the notes are not accepted in other locals, except where they are close by, ports, etc. where exchanges are readily made for their local flavor of currency. I believe this is due mainly to the familiarity of notes and fear of fakes floating about from too many different types.
Different notes from different banks, or issued in different locales makes for neat collecting though! More expensive, sure, but more fun as well.