Synonym for the job title of assistant professor
I think you should just introduce yourself as 


  I work as a professor at the University of X.


omitting the assistant rank that can cause misunderstandings. 

The following conversation line feels also very safe:


  – Where do you work?
  
  – I do research and teach at the University of X.


which might be a better fit since it explains your job in a short sentence. However, nothing wrong with the first line that does not require a question to feel natural.

The actual rank (assistant/associate/full) might matter in a formal (mostly, written and official) context and rarely means a lot outside of the academic system. And that seems to be exactly the situation you happen to encounter issues.

Related questions:


How an assistant professor is called in North America?
Can I call an assistant professor Professor?


Cultural and linguistic differences can play a role, so it might not be as appropriate in certain countries; though, I doubt that would matter at all in a circle that is not familiar with academic ranks at all. Moreover, it is likely that there is no confusion because the rank in the other language is very different.

You might also dig in the direction of "Head of XXX lab" (in case it applies) or "research group supervisor", but I don't think there is a good reason. An assistant professor is a professor and no elaboration is needed in a laid-back unofficial atmosphere.