A doctor's first thought when a young patient like yourself comes in with back pain isn't going to be that you're seeking treatment with opiates. Back pain is common, and in the vast majority of cases, back pain in a young patient is a muscular issue related to strain or injury. With the triggering event being exercising in the gym, no matter how light you think it was, this applies to you too. This is why the GP performed a physical examination, which presumably was unremarkable, and didn't consider it a reason for further evaluation necessitating and x-ray/MRI. The red flags of back pain are: fever, weight-loss, worse pain at night, unusual lumps/pain on palpation, or incontinence. Similarly if you had pain or tingling radiating into your legs that would also be a reason for further evaluation. From your description of the problem, none of these applied to you. I am sorry that the GP didn't explain this to you in proper detail. They didn't mean to make you feel neglected or invalidated.
The GP would never prescribe you opiates for this situation as a 1st line treatment, and you don't have a history of opiate abuse, so your fear likely wouldn't even cross their mind. The pain management she was referring to would have been a combination of over-the-counter pain medication like ibuprofen and maybe a muscle relaxant. GPs have a lot of clinical experience and are usually aware of when a patient is 'simulating' an experience to obtain pharmaceutical relief. I think it's highly unlikely that this was a suspicion held by your GP.
She may, however, have picked up on your anxiety and wanted to discuss your treatment for that reason. I know anxiety can make our fears feel very convincing, but please rest assured that you are not being labelled as a drug addict. You are a person who needs relief, reassurance and better strategies to reduce further strain in the future.