Reporting of RIDDOR applies if anyone (for example, employee, customer of a shop, patient in a hospital) is injured or killed from an accident on Works or commercial premises. So, yes, it POTENTIALLY applies in your case.
However only certain more serious injuries are RIDDOR reportable. RIDDOR reporting enables the HSE to investigate to see whether a breach of the Health & Satety regulations occurred that might lead to the employer being prosecuted - nothing more.
It is still the employer's responsibility to investigate the accident fully and report internally, as well as to you (if you were the patient who was injured). RIDDOR reporting doesn't change the opportunity for insurance claims on the employer/premises owner. However in the event that the HSE decided to prosecute clearly strengthens the likelihood of success of a claim.
No, you can't go contacting the HSE yourself - the organisation should go it if it is necessary. You should ask them why they have categorised the injury in the way they have if you think they got it wrong. The list of reportable injuries is here: -
http://www.hse.gov.uk/riddor/guidance.htm#what