Sim card stores short messages.

Yes, but sim card can only store a small number of short messages (20-50). When the SMS stored in the sim card is full, it will be automatically transferred to the mobile phone. You can change the default storage location in SMS settings.

SIM cards are divided into 8K, 16K and 32K. (written on the back of SIM card)

Now it is generally 16K, 32K.

The higher the value, the more short messages in the SIM phone book.