This may be a tricky one. The gym may be saying that under the terms mentioned in the final sentence you still needed to give notice (presumably still a month's notice) that you wanted the contract to end at the end of the 3 month minimum period.
The final provision may mean that if your 3 month contract finishes on 5 June (say) you'd have had to give notice on 5 May, whereas under a continuous contract you'd have had to give notice on 1st June to finish on 30 June. However the gym doesn't seem to be arguing this now, saying you need to give a full calendar month notice.
On balance, I think someone at the gym is making a mistake here so keep pressing them.