Hi Mark - don't panic! What you possibly have is quite a common injury - you will know more after your MRI. The MRI is just to back up the GP's findings. If you have a damaged disc it means that the disc is bulging out of it's normal boundary and pinching a nerve. This can be caused either by a traumatic event or by repeated straining on the spine. Your question "can I run?" - did you not speak to the Dr about this? TBH - I had traumatic disc injury 3 years ago and there is no way I could even think of running - I actually passed out due the pain the day after I hurt my back when I tried to get out of bed! So, this suggests your injury is less damage!
Have you tried to run? I suspect you will not be able to - it will hurt more and all your muscles will tighten up to protect the injury. What you can do is rest or modify your aerobic activity - try cycling, but there again I couldn't cycle. Swimming possibly? Be careful because if your back muscles go into spasm... believe me - you will know about it. Heat and ice application to the area helps as does lumbar stabilisation (core strength) excersises which do not affect the herniated disc. A physio would help you with this as you do not mention which vertaebra the damaged disc is at.
I don't know how old you are or if the damage is caused by wear and tear on the disc. We all get minor changes to the spine as we age. My daughter damaged a disc earlier this year - she is 24 and was in agony. It took a good 8 weeks to put her right but then she did too much (she is a talented track athlete), came on too strong too soon despite her Mum telling her(!!) and she put herself back to square one at least twice. Five months on she only just starting to fully train again - again I think she was in much more pain than you describe.
When you've had your MRI and discussed it with your Dr - I'd definately get into some core strength work to give your spine all the support you can give it. Both my daughter and I are avid Pilates fans - it's made a huge difference to us both for our running and in day to day life. We tend to take our spines for granted - something I wil never do again!