I don't know how to post questions under your question rather than posting like I for sure know the answer...
I'm not sure if this is helpful, but if the external HDD was formatted by the 64bit Win7, it may have created a GPT Partition (assuming the 64bit is running UEFI rather than legacy BIOS) This might cause a problem for the 32bit computer to be able to read the data if it's running legacy BIOS, which can only read MBR partitions. Does the 32bit system see the HDD as a 3TB drive, or as a 2TB?