No, the “key” (i.e., the Generated Symmetric Key, a.k.a. “account encryption key”) is not generated from the master password — it is generated by a random-number generator.
The Generated Symmetric Key is symmetric, so it is used both to encrypt your secrets into cipher strings stored in your encrypted vault, and to decrypt those cipher strings into plain text. The Generated Symmetric Key is literally the key that unlocks your vault. It should be rotated if there is ever a chance that it has gotten into the wrong hands. This could probably happen several ways; for example, if you have set the vault timeout to “Never”, the Generated Symmetric Key is saved (unencrypted) in persistent storage on your device, so if a bad actor gets access to your device (e.g., an “evil maid” attack), they could easily image your harddrive without your knowledge, and then read the saved Generated Symmetric Key (which, along with your encrypted vault, which is also saved in persistent storage on your device, will provide full access to all of the secret information in your vault). Even if your vault timeout has not been set to “Never”, the Generated Symmetric Key could be written to persistent storage during routine memory management by the operating system, during hibernation, or as part of a memory dump file generated by a crash.
The master password is used to generate a Stretched Master Key, which is also a symmetric key; it is used both to encrypt the Generated Symmetric Key (producing the Protected Symmetric Key), and to obtain the account encryption key (Generated Symmetric Key) by decrypting the Protected Symmetric Key. Thus, in essence, the master password unlocks the account encryption key, which in turn unlocks the vault. An analogy would be that the master password acts like the key to a lock box that holds your vault key (the Generated Symmetric Key). Thus, changing the master password (lock box key) would not affect the vault key (Generated Symmetric Key) or the way that secrets are placed in the vault or taken out of the vault.
Problems occur when an app needs to make a change to an item stored in the vault. After you edit the item in the app, the app then encrypts the secret information (using the Generated Symmetric Key) to produce a cipher string, which is then transmitted to the cloud vault and stored in your vault database in the cloud. However, the Generated Symmetric Key available to the app is the one in the “lock box” that it received when it last logged in. If this Generated Symmetric Key is “stale”, because you have rotated the account encryption key in the web vault (thus generating a new random number for the Generated Symmetric Key), then the cipher string that was just uploaded into your vault by the app cannot be decrypted by the new account key! And when you close your app (unless you have disabled “Lock with master password on restart” or set the timeout to “Never”), you will lose access to the stale account key, so your newly uploaded data is now unrecoverable.
Similar problems will not occur when you just change your master password. Any app still logged in (using the stale master password) can still decrypt and encrypt items using the same account encryption key as before, so uploaded cipher strings can still be decrypted in future sessions. The only difference is that after logging out and logging back in, the app receives a new version of the lock box (which opens with a new key — the master password), but the vault key contained in the lock box is the same key that was previously used to lock up your secrets in the encrypted vault.