Do I have a legal case against the manufactureror against the gym regarding equipment that injured me?

You may have claims for negligence and strict liability against the manufacturer and the gym.

When you complete your medical treatment and are released by the doctor, obtain your medical bills, medical reports, and documentation of wage loss.  Your personal injury claims should include these items and should be filed with the insurance carriers for the manufacturer and the gym.  Compensation for the medical bills is straight reimbursement.  Compensation for wage loss is straight reimbursement.  The medical reports will document the nature and extent of your injuries and will be used to determine compensation for pain and suffering.  Compensation for pain and suffering is an amount in addition to the medical bills. A scar would result in increased compensation for pain and suffering.  If you are dissatisfied with settlement offers from the insurance carriers for the manufacturer and gym, reject the offers and file your lawsuit.  If you settle with one of the parties, then your lawsuit would only name the remaining party with whom you have not settled. If you settle with both parties, NO lawsuit is filed.  If you don't settle with either party, your lawsuit will name both parties as defendants.  If the case is NOT settled, you will need to file your lawsuit prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.

Your lawsuit would have separate causes of action (claims) for negligence and strict liability.  Negligence is failure to exercise due care (that degree of care that a reasonable manufacturer or a reasonable gym would have exercised under the same or similar circumstances to prevent foreseeable harm).

Strict liability is liability imposed regardless of whether or not due care was exercised.

The manufacturer is liable for a defective product.  The gym is liable even if it could not have discovered the product was defective.

Another possibility here is that the product may not have been defective, but may not have been properly maintained by the gym.  This would be an argument the manufacturer would raise.

Another issue is whether warnings or instructions would have prevented your injury.  The manufacturer is liable for failure to provide the warning on the product. 

If the product is defective, is this a design defect or a manufacturing defect?

As mentioned above, negligence is the failure to exercise due care to prevent foreseeable harm.  To prove negligence, one must prove duty (of due care), breach of duty (failure to exercise due care), actual cause, proximate cause, and damages.  Actual cause means but for the band snapping would you have been injured?  If the answer is no, which appears to be the case, actual cause has been established.  Proximate cause means were there any unforeseeable, intervening events which would relieve the manufacturer and/or gym of liability?  If the answer is no, proximate cause has been established.  Damages means the amount you are seeking to recover in your lawsuit.  Damages would include your medical bills, pain and suffering and wage loss.