I watched the video and it maybe that she was on public land ... where does your property actually start? Isn't there part of the land that is public between the street and your house? Video is hard to tell. Take some pics and post...from different angles. And it certainly looks like you invite people over to where the cans are...
No one wants anyone snooping through their trash for whatever reason.
As to what you can do .... consult google scholar and do some R&D -- different states have different rules.
Why you telling the cops you had a gun? Not smart. Don't care if you have carry license, in fact, it should make you understand not to tell.
You have a garbage picker ... next time hand her a notice of trespass (if this has any legal effect, chk your state laws~generally a phone call is not sufficient); if it has legal effect then you may arrest her if she actually trespasses (again consult your state/local law). But beware .. the person you detain could turn around and sue you...even if you have the law on your side.
Solution maybe to buy a bin for what she is searching for (metals most likely) .. then she can collect it easily enough.