As an insurance broker, the difference between the two policies is, the Public Liability covers you for third party accidental bodily injury or damage to property caused by your negligence. The Professional Indemnity covers you as a professional coach for any bodily injury etc caused from your teaching.
Regarding your query about someone slipping or falling on the stairs, this has nothing to do with your coaching and should be covered under a Public Liability policy. If the injured person was a friend of the house owner and was there as a guest that would be covered under the Liability section of their House insurance. However there are always exclusions. Running around a pool area where they may slip, or just because you fall whilst going up or down the steps is their own fault and should not be covered.
As you are holding classes, you should ensure the work place is in a safe condition before you use it. You should have your own Liability policy to cover any accidental bodily injury. You leave a bag somewhere that could be tripped over and a toddler gets hurt, that is your negligence for not putting your bag away in a safe place.
I would suggest most insurers are not interested in this type of business anyway, but as it is being held at a "friends" house does not help either. If you were holding classes at a recognised club, they should have all the necessities in case of any accidents. Lifeguards that know CPR, medical kits for injuries, easier access for ambulances etc.