(A versus the) background check

Hi. (Don't be put out by the length of the post, the Q is easy!)

I was taught that once you mentioned a noun, all subsequent mention of should be preceded "the".

Now I know it's not always true.

Here are two situations. I am the employer and I am hiring a new person, Bill.

Situation 1.
My assistant: "Boss, we need to do a background check on Bill."
Me: "Okay, go ahead and do a background check."
A few days later I ask my assistant:
Me: "So, did you do a background check on Bill?"
(My assistant is forgetful and no background check might have been started for all I know. It's still "so did you do ANY background check on Bill"?)

Situation 2.
Me: "Bill, we will do a background check on you, OK?"
Bill: "Yes, no problem."
Then I sent Bill an e-mail:
"Bill, this is just to confirm that we've arranged for a background check on you as discussed."

I think "a" is appropriate as I put in bold. At all of these points, this background check is still general (not specific enough). It's still "any background check".

Does this make sense? I am not sure if I have expressed myself well enough.

And thank you.