Regardless considering what had happened I think this was pretty close to an emergency without actually saying it.
Of course it was an emergency. However it may be worthwhile to do a full debrief with your instructor during which you may realise that the situation, though serious, was at all times under control by virtue of his/her skills acquired during training.
You and your instructor were faced with, at best, an unreliable engine. Happily in your circumstance, it would appear you had sufficient height to return to the airfield.
Moreover, your instructor did a fine job in evaluating the situation and determining that a landing on 29 would be a better option.
You dont tell us whether he pulled full flap on short final once absoutely sure of making the runway, and if not, this may account for the high approach speed. Either way a landing at the runway mid point would indicate that he/she used good judgement in determining that there was sufficient runway to land on and stop safely.
It is most unfortunate that you experienced this situation so early in your training, however this is exactly the type of scenario that you will rehearse and simulate should you decide to proceed with your PPL. The fact that your instructor practised this during training should be sufficient evidence to you that the training works and is designed to equip you with the skills necessary to cope.
One of the most gratifying parts of my training was the opportunity to gradually develop and subsequently hone my abilities to make good judgement calls which, inside or outside aviation, is a useful skillset to possess.
My advice would be to treat this as a valuable experience and aim to acquire the skills and judgement which your instructor has demonstrated to you for real.
Safe flying