General aviation pilot reported mis-communication with ATC after receiving an altitude and heading assignment on an IFR departure. The communication issues resulted in an altitude deviation and altitude alert warning from ATC; then the pilot returned to the cleared course.

Narrative

Situation: Altitude deviation on IFR departure due to task saturation induced by miscommunicationOn an IFR departure; I was assigned an altitude to climb to (2100 MSL) and a heading (120) to maintain until given radar vectors to the next fix (a VOR) by ATC. After being handed off departure from tower; I checked in with ATC with the tail number; my current altitude and informed them that I was climbing. To keep the radio call short I did not state which altitude I was climbing to as that was part of the previous clearance. My first radio call was missed by ATC. Later; ATC called with 'XXX are you up?' I answered immediately with the tail number XXX; my altitude (now higher than on previous call) and that I was climbing. I was then informed of 'radar contact'; an altimeter setting and 'verify you're assigned altitude'. What I did hear was 'verify your altitude'. This seemed strange to me as I was still climbing to the altitude I was given. I communicated my new altitude on ATC. I started to wonder if there is something wrong with my altitude. ATC came back with 'altitude should be 2100'. I then confirmed; XXX is climbing to 2100. Quite some time has now elapsed since my first radio call and the confirmation to climb to 2100; which was already part of my clearance. At this point I just finished turning to the initial clearance heading of 120. The next communication I heard was ''negative turn; you should be turning on course'. I did not understand if I heard negative turn as it's a terminology I had not heard before and I was already on a 120 heading course for sometime. Did this mean turn back? turn the opposite direction? I assumed it meant 'immediate turn; you should be turning on course'. I complied; turning immediately to my next point; direct ZZZ VOR; which is a left turn and also complying with the opposite direction. I was surprised by the confusion in the turn and the urgency to make a turn in the opposite direction before reaching 2100. I assumed this was due to the very long time elapsed in the previous communication calls and that the instruction to turn was not communicated to me yet. To accelerate the immediate turn request; I disconnected the A/P; which was on a 120 heading; and made a sharp left turn. Inadvertently I lost altitude in the process. Confused; while in the turn I told ATC 'Aircraft X is turning to ZZZ; do you want me to change?'. I heard back from ATC no information about the turn; but rather 'altitude alert; climb to 2100'. This is when I realized that I lost some altitude during the turn; I then climbed continuing direct to ZZZ as there was no heading correction; and later was given next vectors out of the area. There was never any call out about the heading change to the ZZZ VOR; which confirmed me that I properly complied to that instruction. There was miscommunication on both sides. The controller may have not been hearing me properly and was trying to get me to climb to 2100. With my utmost desire to comply; I perceived an unusual sense of urgency specifically for the turn to my next fix and not as much for the climb (immediate versus expedite instructions). I started to turn direct to it without vectors and asked for verification. This contributed to task overload close to the ground and I lost some of my altitude of climb; which was then rectified as soon as I saw it. The total length of this event; which was about a minute on the radio while close to the ground certainly increased the risk. This situation can be improved by:1. ATC to use a more concise language. This would reduce the communication time and; enable me to better focus and execute the request.For example: instead of asking me to 'verify assigned' altitude; I could be told just 'climb to 2100' from the first try. Using the language 'climb IMMEDIATELY to 2100' would have helped me realize the urgency for me to climb to 2100 from my previously assigned altitude by the airport tower instead of 'expedite' in subsequentcalls. It would have not taken us over 30 seconds to be on the same page.2. Receive clear instructions for turns from ATCFor example: Turn RIGHT heading XXX or Turn LEFT Heading YYY. Use of words should as negative turn are harder to decipher and I thought it meant immediate turn. 3. In hindsight I should have verified the turn request before starting the turn. This would have helped me avoid task saturation leading to involuntary altitude loss and would have achieved the altitude request faster versus introducing a new variable in the flight (a turn). When load increases in the cockpit; do not disconnect the A/P and leverage automation to reduce pilot workload.