The process used to migrate the HDD to the existing Intel SSD 240 GB should be the same that you used in the Lenovo PC to migrate the data in the old SSD to the new SSD. From your explanation, the SSD from Lenovo is detected properly by IDMS as an Intel drive, otherwise it would not allow you to start the migration process, or even install the application.
It is advised to do a Secure Erase of the target SSD before initiating the migration process, or at least deleting the data and volumes from it. However, this is not a reason for whirring sounds.
As I understand, the HDD in the PC is making sounds during the migration, and eventually the clone operation fails. If this is the case, then the old Hard disk may be failing. If there are errors reading the source drive, then the cloning process will fail. Intel® Data Migration Software has to clone All the data in the drive; including partition information and files; if some portions of the data cannot be read correctly because the source has bad sectors or corrupted data, then it will not be possible to do a full clone onto a target drive.
If this is the situation you are facing, the first thing we advise you to do is a manual backup of any important folders and files from the failing HDD to a good drive.
Take into consideration that there if there are unreadable files in the old drive, you might need to manually select the good ones.