Nope, president and prime minister is not the same thing.
A prime minister is always the head of an administation and nowadays is responsible to the legislature - even if theoretically responsible to the monarch.
Presidents are directly elected and responsible to the electorate.
That isn't really the case.
The Monarch's useable power has diminished over the years, but it was well after Magna Carta - and the reserve powers that still exist today and could be used in an emergency are still very real. The actual Monarch hasn't used their reserve powers in a long time, but in my country (Australia) the Queen's representative, the Governor-General last used them as recently as 1975 - he sacked the entire government to resolve a constitutional crisis.
More significantly, ultimate power to command the military rests with the Queen. The emergency orders that would release nuclear weapons in the event of an attack of the United Kingdom are 'Queens Orders'. Yes, the Prime Minister has the power to issue them, but they are the Queen's orders.
And if the Prime Minister is killed or incapacitated, there is no automatic succession. A new Prime Minister must be commissioned by the Monarch. However, if the Monarch is killed, the next person in the line of succession instantly becomes King or Queen. For this reason, maintaining the Monarchy in the event of an emergency is more critical to British survival than maintaining the Prime Minister. The King or Queen (whoever that may be) can govern alone if necessary. A Prime Minister cannot.