Your confusion makes sense considering there are three different forms of continuation applications: 1) (straight) continuation, 2) continuation-in-part (CIP), and 3) divisional. In your case, it sounds that your assignee is seeking a continuation or a continuation-in-part.
A continuation application uses the same specification and drawings of its prior parent application, but adds new claims or new amendments to its prior application that changes the scope of the parent application. A continuation application marks back to the original filing date of the parent application. Also, there is no limit on the number of continuation applications as long as the parent application is still pending. Side note - the USPTO attempted to limit the number of continuations in 2007 but was unsuccessful after it received heavy opposition against the proposed rule change (so be mindful of a potential revision).
A continuation-in-part application is a tool utilized by fiscally-sound inventors as it allows you to supplement your existing patent without going through the burden of filing from scratch. However, if approved, the addition(s) does not go back to the original filing date and instead retains a filing date for when the CIP application was submitted. Similar to continuation applications, the USPTO does not limit the number of CIP applications.
A divisional application is filed typically after a USPTO examiner determines that a patent application contains more than one invention. The examiner will issue a “restriction requirement,” compelling the applicant to choose which invention the examiner will search and examine in the parent application. Thereafter, the applicant will submit a divisional application to specify which invention he/she intends to move forward with.