Unfortunately as that English Language and Usage question mentions the terms résumé and CV have, to a degree, come to be used interchangeably in the US.
Generally the only places I've encountered that truly want a CV are academic institutions (interviewing for professors/lecturers/etc.) and law firms (big cases you've argued), but your region (or prospective employer) may vary.
Bottom line? If you have any doubt whether they want a summary résumé or a full CV, ask.
Submitting a 10-page CV to a company that really only wants a simple one-page summary of your work experience will probably hurt your application, but a polite inquiry to the HR person ("I see you've requested my CV - just to clarify do you want the full 10 pages, or a one-page summary?") can avoid the whole problem.
There's also nothing wrong with having "Full Curriculum Vitae available at
URL" at the bottom of your résumé.