What you tell the lab is a separate issue.
You may rate the stock at something different than the manufacturer's suggested rating, for whatever reason. For example, you may want to rate a 500 ASA stock at 400 ASA to slightly overexpose it.
Normally you would still just tell the lab "PROCESS NORMAL." Then the processed negative would end up 1/3 of a stop more dense (overexposed) than if you had rated the stock at 500 ASA. You then would probably correct that slight overexposure when making a print (by printing "down", aka using a higher set of printer light numbers) or darkening it in the telecine transfer. It helps, if you aren't there for the transfer, to shoot a grey scale or card at the rating you have chosen so that the colorist will attempt to make the card look "normal" in brightness and color, and thus correct for the extra density of the negative.
But you may tell at lab to PUSH or PULL PROCESS the negative by "x" number of stops. And you may have exposed the film to compensate for this, like rating a 500 ASA stock at 1000 ASA and then asking the lab to PUSH ONE-STOP and thus compensate for the underexposure by increasing the neg density through extended development, thus hopefully ending up with a normal density.
You could even rate a 500 ASA stock at 800 ASA, for example, which is a 2/3 stop underexposure, and then ask the lab to push one-stop and have a final negative that is 1/3 of a stop denser than normal.
The lab doesn't really care which color neg stock you give it or how you exposed it -- it all goes into the same ECN2 process. The only thing that they care about is whether you want them to process normal, or pull or push the process (or special processing like skip-bleach.)
Any final adjustments to scene density after that would probably be done in making the workprint or in the telecine transfer.