Why must the definition of enthalpy include a reference to "constant pressure?"?

If we allow for constant volume (which means the pressure can change), we measure the change in the internal energy (symbol = ΔU or ΔE). It has been determined by chemical minds MUCH better than my own that changes in enthalpy (the constant pressure value that is measured) is more chemically interesting (and useful) than measuring the change in the total internal energy.

It may be as simple as using a calorimeter open to the atmosphere (a constant pressure calorimeter) is a more convenient tool than a constant volume calorimeter, because the CV calorimeter must be sealed and, if it loses its seal, the results are rendered useless. Using a CP calorimeter is much, much simpler and cheaper.