I don't have much experience with Service Desk, but in JIRA Core/Software, there a couple of default user fields: Reporter and Creator. The Reporter field captures the person who reports the issue, while Creator captures who actually created the issue. To my knowledge, the Creator field is only available on the Issue Navigator as a column and cannot be added to any screens (including the Issue View screen).
In your case, you could manually set the Reporter to the customer to open the issue on their behalf. You would be set as the Creator no matter what, but the customer would be the Reporter. You could then view this in the Issue Navigator to see who created the issue on the customer's behalf.
Again, I haven't really used Service Desk, but hopefully this can point you in the right direction.
It's not the most ideal method, but this information should already be saved for you if you need to go find it. You could go a step further and created a scripted field that returns true/false to let you know if the Reporter = Creator, or create a scripted post-function to add a comment after creation if the Reporter != Creator, but that would require an add-on if you don't have it already.
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