Integra 96 gsr wont crank.

I have a 96 integra gsr vtec 2dr b18c1 engine stick shift with aprox 150,000 miles on it. And its not cranking. When I turn the key, my dash lights are dim and I have tested the battery on another car that works fine. Check engine doesnt come on and a weird sound comes from the abs and a clicking sound. I have tried to push start the car and it starts fine. I also have checked the battery, starter and tested the alternator at the shop and it was fine. I think my starter is not getting enough power to start it up. I have a battery jumper and when I try to connect the positive to the starter and the ground to the engine. Then turn the key to the on, it looks like its getting alot of power as the check engine light would come on and go away then starts up with no problem. Already check the fuses and they are ok. Any help or suggestion will help. Thanks.