L1 appears to be a toroid (?), i.e. for all practical purposes it has a closed magnetic path - and therefore very little interaction with an external field.
Yes. Your "detector coil" could not be any worse.
Imagine you have a slinky. The magnetic path runs down the middle of the slinky. If you have a slinky with nose to tail (like a donut), the magnetic path runs around in a circle, not interacting with the outside world very much.
What you need is a slinky just sitting flat - the way it came in the box - with two open ends. The magnetic path runs through the middle of the slinky (just as before) but now has to turn around and wrap outside the slinky to get back to the other end - and this hugely interacts with the outside world.
In short, you need a flat coil (a very short slinky).