Summarising all the comments, the situation seems to be this.
On your CV you have quoted 7 years of employment with your current employer - possibly with 2 years attributed to "internship" (this point is unclear).
In fact you have only 5 years of ordinary employment with your current employer, with the remainder of the time concerning "project work" in collaboration with them. This "project work" was arranged by your university, but not officially called "internship".
You also say you had a role with the university itself as a "teaching assistant".
My first question would be how far different (in terms of time consumed or experience gained) is this "project" work from an internship as commonly understood. And was it reasonably continuous over the 2 years?
Also, is your "teaching assistant" role also clearly shown, and your Master's course?
If the "project work" was spread over most of the 2 years, accounts for 2 years of your time in a substantial way, and if a person could reasonably see that the same span of time was claimed by three different activities (Master's course, teaching assistant, and intern), then I might be inclined to think it corresponds with the truth.
Where I might think it doesn't correspond with the truth, is if the "project work" turned out to be quite trivial, or if the "teaching assistant" role were not shown at all (even though this was, by your own account, one of the main engagements).
If you're being asked to fill in a form for a background check, the easiest thing to do is use the opportunity to explain what the entry on the CV represents, and ask the question about how it should now be represented on the particular form.
It's quite reasonable that a CV may show your working with an organisation, whereas a background check might want to understand who the legal relations were with, who kept the records of the relationship, and how would those records be filed by the organisation, with a view to soliciting verifying information in an effective way.
If you have really deviated from simplifications and "seller's puff" into what many would regard as somewhat of a falsehood, then perhaps the only rescue is to pre-empt the discovery of the falsehood, and explain that you've had new feedback on the matter and you think there's a potential for readers to be misled.
Finally, if you have deviated into what most would regard as obviously mendacious statements, then it may be best to simply withdraw the application and stick with your current employer for a while longer.