However, if you load up your torrent manager and say "download please!" you are making your own copy, which is then stored locally, just like pushing the button on a copy machine.
Only if you actually made a durable copy of the file, and they won't have any evidence of that from the network traffic. All they know is that someone else sent a copy of the file to you. That would support a case against the uploader, but not the downloader. It might be enough to get a warrant to examine the downloader's device for a stored copy of the file, but it's unlikely anyone would go to the effort of actually serving a warrant to recover, at most, a small multiple of the retail value of one copy of a single work, and until they do so there is nothing to support a charge of copyright infringement.
Of course the root of the problem is copyright. This is just one of the more notably absurd, and yet inevitable, consequences of trying to impose artificial scarcity on something that can be duplicated by anyone at effectively zero cost.