Rushes have round blades in cross section so that would be an easy thing to check. Does it? If not, carex is something to look at too. Rushes are round, sedges have edges.
Here is not such a good picture of blue-eyed grass in my yard with some seed pods for comparison. I can't really see good detail in your photo. Right now, those pods are brown here, but I could take new photos. It blooms in May/June. It is the species, not Lucerne. I also have juncus growing in my lawn. The seeds look more like dried flowers, not round pods. I can't think of the right term now. Of course there are many different rushes.
Here are pictures of Sisyrinchium angustifolium on the Prairie Moon website
https://www.prairiemoon.com/si...
There are many sedges. If you can get a good photo of the seed pod and how it is on the stem, that would help a bit.