When your Bitwarden client is logged in and locked, then your stored secrets exist on your device only in an encrypted form, within a locally cached copy of your vault. No there is nothing wrong about keeping Bitwarden logged in but locked - this is they way Bitwarden is meant to be used, and I believe that most Bitwarden users follow this practice. Alternatively, if you have a Premium subscription, you also have the option of using the Bitwarden Authenticator feature to generate TOTP codes directly inside the Bitwarden password manager. In that case, if you have a free Bitwarden account, you will need a third-party TOTP authenticator app (e.g., Aegis, Raivo, 2FAS, or Yubico Authenticator) to generate the TOTP codes. With regards to Application #2, the 2FA methods possible for logging in to your stored accounts depends on what is offered by those websites, but let's assume that you are mostly interested in setting up 2FA using TOTP codes. If you have a Premium subscription, you also have the option of using Duo, YubiKey OTP, or FIDO2/WebAuthn as the 2FA method for your Bitwarden login.

With regards to Application #1, if you have a free Bitwarden account, you can do 2FA for your Bitwarden login using either a one-time code sent by email, or using a TOTP code generated by a third-party TOTP authenticator app (e.g., Aegis, Raivo, 2FAS, or Yubico Authenticator). 2FA to log in to the accounts that are stored in your Bitwarden vault.2FA to log in to your Bitwarden account.

You are confusing two applications of 2FA:

Bitwarden vault login