When dealing with different regulations, one has to be careful about definitions. For instance, the definition of a customer for CIP is different than the "general" definition of customer in FinCEN regulations.
FinCEN - General Provisions
Sec. 1010.100 General definitions
When used in this chapter and in forms prescribed under this chapter, where not otherwise distinctly expressed or manifestly incompatible with the intent thereof, terms shall have the meanings ascribed in this subpart. Terms applicable to a particular type of financial institution or specific part or subpart of this chapter are located in that part or subpart. Terms may have different meanings in different parts or subparts.
(p) Established customer. A person with an account with the financial institution, including a loan account or deposit or other asset account, or a person with respect to which the financial institution has obtained and maintains on file the person’s name and address, as well as taxpayer identification number (e.g., social security or employer identification number) or, if none, alien identification number or passport number and country of issuance, and to which the financial institution provides financial services relying on that information.
FinCen - Rules for Banks
Sec. 1020.100 Definitions.
Refer to §1010.100 of this Chapter for general definitions not noted herein. To the extent there is a differing definition in §1010.100 of this Chapter, the definition in this Section is what applies to Part 1020. Unless otherwise indicated, for purposes of this Part:
(c) Customer. For the purposes of §1020.220: (aka - Customer Identification Program-CIP)
(1) Customer means:
(i) A person that opens a new account; and
(ii) An individual who opens a new account for:
(A) An individual who lacks legal capacity, such as a minor; or
(B) An entity that is not a legal person, such as a civic club.
(2) Customer does not include:
(i) A financial institution regulated by a Federal functional regulator or a bank regulated by a State bank regulator;
(ii) A person described in §1020.315(b)(2) through (b)(4); or
(iii) A person that has an existing account with the bank, provided that the bank has a reasonable belief that it knows the true identity of the person.
For purposes of requiring your CIP process, the definition of a customer is different than it is for all other areas of BSA. An item to consider would be to define that in your Bank's policy using the "general" definition for anything other than your CIP process. For instance, if you are providing financial services and are relying on information that you have on file in providing those services, then that person is your customer.
In the case of cashing a check for a "former" customer, this is where your policy and procedure should be clear to front-line staff - are you relying on old information you still have in your system to cash the check? Or do you require the person to present a current form of identification before cashing the check?