I don't know where you are getting your information from. If its from the police who arrested you, think about how truthful they have been.
Yes, the police will keep a record of your arrests, but only convictions go on the public record. So a background check is NOT going to find that you were arrested.
Second, we are only hearing your side of the story here. We don't know what the circumstances were that you were picked up in the first place. We don't know what proofs or evidence you were able to produce that showed you were not the same person the warrant was for. We don't know what charges the warrant was for.
This is why we are insisting that you speak, personally, to an attorney. You will probably not be charged for an initial consultation. If you were as wronged as you say you were, then its probable that you can find an attorney to take your case on a contingency basis, which means you only pay if you win.
While there are some attorneys who post here, many of us, including some who have responded to your plea, are very familiar with the justice process. No attorney is going to be able to give you accurate advice based on the little you have told us. So, again, contact an attorney.