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Try cycling the battery a few times this might fix the issue.
If this doesn't help either ask for a replacement/refund from the seller or just use it as is.
If you keep using it you should do some benchmarks (Geekbench etc.) Because the CPU might be throttled. Some MacBook do that when the battery is defective/not installed.
Yes there are multiple machines with the model identifier A1278. In fact all non retina 13“ MacBook Pro models share the same A1278 model identifier as well as the Late 2008 Aluminum MacBook. However only the Late 2008 MacBook has the split access door.
All Non-Retina 13" MacBook Pros and the Late 2008 Aluminum MacBook have the same A1278 Model Number (I guess due to the almost identical chassis).