At equilibrium (that is, if volume/temperature/mass is stable and not changing over time), the external pressure applied to the gas will equal the internal pressure of the gas.
This is (more or less) a consequence of Newton's third law. The external environment pushes on the gas with a certain force, and the gas pushes back with an equal and opposite force. At equilibrium, no work is being done on the gas or on the environment, so the forces have to balance. (This is very crude and hand-wavy - a more thorough description would require much more math.)
So (at equilibrium), there really isn't any difference between the two. In the Boyle's law case, the value is described as the external pressure, because the typical experimental setup in that case involves using something like a piston to control and vary the external pressure applied to the system. In the ideal gas case, you may not explicitly be controlling the external pressure, instead you're controlling temperature, volume, or particle numbers. In those case, you're normally measuring the internal pressure of the system, rather than (directly) controlling it or the external pressure. But (at equilibrium) the pressure of the gas pressing out is the same as the pressure of the container pushing in.
It's all just a matter of perspective, and how the system is set up.