That guidance is a bit "simplified." There is an old thread on Batteries under Non-Food items that includes a link to more detailed information on what is and is not prohibited:
b. Batteries. Mailable batteries include:
(1) 49 CFR identifies lithium batteries as Class 9 (Miscellaneous Hazardous Materials). Primary (non-rechargeable) lithium batteries and cells are prohibited via air transportation. Mailable batteries must be securely packaged to preclude a direct short. A strong outer packaging must be capable of firmly and securely holding the inner contents and must be clearly marked on the address side with "Surface Mail" or "Surface Mail Only." In addition, the general packaging requirements in DMM 601.1 through 601.8 apply.
(2) Common household dry-cell batteries such as sizes AA, AAA, C, D, etc. are generally not regulated as hazardous materials and therefore mailable. Mailable batteries must be removed from the device they power and protected from short circuiting. Packaging requirements in DMM 601.1 through 601.8 apply.
(3) A nonspillable wet battery containing liquid electrolyte is permitted to be mailed only if the battery casing is completely sealed to prevent the liquid corrosive from spilling during Postal Service handling, as stated in DMM 601.10.19.4. Nonspillable batteries with UN2800 are prohibited in international mail, but may be sent as domestic mail via air or surface transportation when prepared within the conditions of Packaging Instruction 8B in Appendix C.