My native language is Persian (Farsi). There is no definite article in Persian and the specific object/ person/ idea which a noun refers to is determined from the context.
I agree with you about the ambiguity of the word "singularity". Not only there are different definitions for "singularity" in AI, the term is also applied in other contexts (e.g. economic singularity, gravitational singularity). I think, as you said, talking about "a singularity" is more appropriate.