I recently have swapped my headlight casings and all the oem parts besides a ballast swap I did in the left one. It came with a new ignitor lamp holder, but also came with an ignitor wire which I've seen clips on to other cars as well as later gen TLs, but I see no copy of this in the OEM one or on 2001-2003. I want to make sure that I am not running open current for a connection that I don't have. Is it fine to leave it, should I cut it off?

 

picture please? I'm not sure what you're referring to

 

Here is what they both look like:
(side note I finished my installation, they both work fine)

This is the OEM one ^

This is the aftermarket eBay one ^

 

Strange, might just be an extra ground cable on some headlights.

 

The wire appears to be soldered to the metal case, if that is true it is a ground connection. If so you can connect this to any metal part of the car chassis. As Thoiboi mentioned these are used on multiple different cars so sometimes you wind up with "extra" connections. I usually connect extra ground connections to the chassis.

 

These are two photos from a similar configuration from a 3rd gen tl that is similar to the ignitor i put in my 2nd gen tl. It appears there is no bar to connect it to or the two pin to plug it into in mine, so I left it hanging. Headlights turn on all good.