You nead to remove the SPN that isn't used. As you've found that SQL is running as Local System and is by that reason using SPN on computer account, you shall remove the SPN from Administrator account.
To increase security, you should create a dedicated service account instead of running SQL as LocalSystem or Administrator (both gives to much permissions).
Instead of using ADSIedit, you can use setspn-command to delete the SPNs.
C:\>setspn -D MSSQLSvc/lpz-tickets.LowryParkZoo.com:1433 Administrator