Hi Hans
I think you are correct in thinking that if the experiments where just trying to detect 'dark matter' in general, then they would almost certainly fail.
However, that is not what they are trying to do.
First they make a hypothesis about what the properties might be, based on the very little information that is known.
They basically make guesses about how this hypothetical dark matter might interact with normal matter. They then design and build a detector for that interaction.
This is what the LUX experiment is trying to do. Hopefully it detects something. But, even if it does detect something that is in the range that is expected, this is still only evidence, not proof.
Take a look at this link for this experiment.
http://lux.brown.edu/LUX_dark_matter/Experiment.html
I think the Stawell experiments hopes to make detections at the same time as some previous seasonally related, northern hemishere detections that appear seasonaly.
If they do make detections at the same 'time' they will be at the opposite 'season' indicating that whatever is the cause is not on the earth.
Wikipedia has what seems to be a good article on dark matter history and detection.
http://en.wikipedia.org/wiki/Dark_matter
Enjoy your reading!
Philip