Integrations.type migration: 2 - Backfill `type_new` for all records

About

This issue is part of &6177 (closed). See that Epic for a broader description of the feature.

We want to copy data from type to type_new. The value should be changed in the same way as #333506 (closed), from "<integration name>Service" (their old model names) to "Integrations::<integration name>" (their new and existing model names).

We may be able to make use of CopyColumnUsingBackgroundMigrationJob - see &6177 (comment 600106425).