yes i think it should be chemical energy (or enthalpy to be more precise). enthalpy takes into account kinetic energy, potential energy, and energy required to make room for the system. so, in order to 'activate' a reaction, you need to input enough energy into the system from the surroundings so that the ENTHALPY of the system is above or equal to the activation energy for the reaction. i think the diagram in the textbook, depicting the probability density function of the maxwell-boltzmann distribution is a bit misleading, although it is certainly true that if kinetic energy = activation energy, then the reaction will take place (the enthalpy will then be greater than or equal to the activation energy).
and to answer your second question explicitly: if the enthalpy of a system equals to the activation energy of the reaction that is to occur, then the reaction will take place. the enthalpy does not need to be greater than the activation energy. this is kind of implied in the definition of activation energy: "the MINIMUM energy required to break bonds in reactants so as to allow reaction to commence". if you hit the minimum, then the reaction will occur.