Does at-rest mean: encrypted on storage so noone can physically steal a drive or encrypted in the database so noone can get the information with SQL without the key (e.g. Postgres column encryption)?
Conceptually, yes. You can encrypt at the database/filesystem level (where the OS/DB engine manages the encryption keys and enforces access control), at a table level/column level (where the db engine enforces access control) or at the application level (where the application manages the encryption keys and they are separate from the database engine).
They serve different purposes. For eg: When a disk drive is faulty and thrown away, you may not want data to be recoverable from it. So, the filesystem level encryption helps there. A db/table/column level encryption helps when there are different applications (eg: transaction processing and analytics) accessing a shared database. Reporting queries may not need access to the sensitive fields whereas certain transaction processes may need it. In this case, db/table/column level encryption helps. When you want separation of concerns, you can add application level encryption (on top of the other two). Example: Your data is stored on the cloud and you don't want the cloud service provider to know the data or if they replace a disk drive as part of normal servicing, you don't want your data to leak.
They serve different purposes. For eg: When a disk drive is faulty and thrown away, you may not want data to be recoverable from it. So, the filesystem level encryption helps there. A db/table/column level encryption helps when there are different applications (eg: transaction processing and analytics) accessing a shared database. Reporting queries may not need access to the sensitive fields whereas certain transaction processes may need it. In this case, db/table/column level encryption helps. When you want separation of concerns, you can add application level encryption (on top of the other two). Example: Your data is stored on the cloud and you don't want the cloud service provider to know the data or if they replace a disk drive as part of normal servicing, you don't want your data to leak.
This depends on the threat model.