PCT Controller described operational deviation when aircraft enter adjacent facilities airspace without coordination; alleging flight crew non-standard phraseology read back caused event.

Narrative

While working a secondary departure sector; I failed to catch an altitude report by a pilot and subsequently the aircraft violated the airspace of another facility. During the investigation; it was revealed that the pilot had been assigned 170 but the last instruction the previous Controller issued him was 'Fly heading 330 degrees.' The pilot's readback was '330.' When the pilot checked in with me; he stated 'aircraft X with you 13 for 33;' I failed to catch the abbreviated altitudes he read me. The aircraft then climbed through my altitudes resulting in an operational deviation. 3 things about this deviation bother me. 1) The pilot's sloppy readback to the first Controller '330;' 2) The pilot's sloppy check-in with me '13 for 33;' and 3) Why aren't pilots held to a standard of phraseology that would help prevent hearback/readback incidents. Had the pilot checked in with some semblance of standard phraseology and stated the word 'climb' or 'flight levels' it would have caught my attention and clued me in to listen; 'aircraft X with you 13 thousand climbing to Flight Level 330.'