What the claim of type http://schemas.xmlsoap.org/ws/2005/05/identity/claims/nameidentifier
should be used for?
This is the main question, and here are
Per The Role of Claims,
Name The unique name of the user
Name Identifier The SAML name identifier of the user
These two claims are part of the group of claims that AD FS 2.0 configures by default.
This implies that they are IP scoped.
e.g. when you log in to Google using ACS, "nameidentifier" is the unique GUID associated with your account by Google whereas name is your Google login e.g. "tim.smith@gmail.com".