Fidelity in terms of recordings, records, etc means "the degree of exactness with which something is copied or reproduced."
So what they mean when they say these tests are for "fidelity" is that they are done to determine whether a copy performs the same way as the original that a copy is being made from. In Delos' case its being done to see if the Delos brain in a robot body says and does the same thing the original Delos did. In Bernard/Delores case, the conversations are to determine if Bernard is a faithful copy of Arnold. In the very last Bernard/Delores scene, presumably this is a new Bernard being test for fidelity against what Delores remembers of the last Bernard (not Arnold), and in the final scene presumably a robot copy of William is being tested for fidelity against the recorded actions of the original William, who was presumably a human.
So what they mean when they say these tests are for "fidelity" is that they are done to determine whether a copy performs the same way as the original that a copy is being made from. In Delos' case its being done to see if the Delos brain in a robot body says and does the same thing the original Delos did. In Bernard/Delores case, the conversations are to determine if Bernard is a faithful copy of Arnold. In the very last Bernard/Delores scene, presumably this is a new Bernard being test for fidelity against what Delores remembers of the last Bernard (not Arnold), and in the final scene presumably a robot copy of William is being tested for fidelity against the recorded actions of the original William, who was presumably a human.