So checking the stroller at the ticket counter may or may not work. If that doesn't work, the second option would be trying to check it at the gate. (I'd rather not do that because I have connections in both DTW and AMS, and I'd have to log it wound in both airports.) Between the two, let's say there is a ~75% chance. If that fails though, then I have pay $200 for checking it. If I put it in a box, wouldn't that automatically make it a piece of luggage? Also, if it is in a box, it makes it even harder for me to deal with it in DTW and AMS.

Whether DL makes me pay the $200 or not, it may still get damaged. As GUWonder correctly points out, this may be quite a problem.

Given the risks involved, it makes more sense to mail it. It's cheaper than $200 and I can have it ensured for damage.