Ideally if you could design the nozzle to match the exhaust pressure in a vacuum (i.e. nearly zero), the third term drops automatically.
If $p_0$ is zero, then $p_e$ would have to go to zero as well because an ideally designed nozzle results in no pressure drag (i.e. ambient freestream pressure and exhaust pressure are the same). In reality, such a nozzle could never be built because it would be infinite in length (it takes an infinite length to drop the exhaust pressure to an infinitely small pressure such as a vacuum). But real nozzles are designed to make that exhaust pressure as close to ambient as possible given the constraints of its length while still accelerating the exhaust gas as fast as possible. If I recall correctly, nozzles tend to be optimized for use at ambient pressure close to the launch site on the surface of the earth (because its so hard to lift off), so it makes sense that these terms tend to be included when discussing rocket design.
Also, the freestream velocity of a vaccum would be zero, which would drop the second term. Although, technically speaking, freestream velocity not really well defined. In a vacuum, there is no freestream of anything to begin with, so you can neglect it. The general thrust equation applies more to the case of the presence of a fluid (i.e. air). In a vacuum, those terms just don’t make any sense.
Edit: i dug a little deeper into the meaning of these equations and found that the second term is called the ram drag, which only applies to air-breathing engines like jets. It would have to be dropped for rocket engines because they carry their own fuel/oxidizers. They don’t take air into the engine as part of the combustion process.
So, the second term could be interpreted as a mass flowrate of the intake of air. That flowrate would of course be zero in a vacuum.