Sometimes it is hard to tell what is damaging plants unless you see it in action.  Though I think your theory about birds could be right.  Rabbits would eat all of the plant, but sometimes birds peck at plants and don't eat them all.  They're sampling to see if the want to eat them.  Sometimes they'll eat some and just damage others.  You may want to put some sort of protective cover over the plants until they're a bit bigger and don't look like an easy bite to the birds.  Using mesh, chicken wire, or row cover material should work.

One other thing to check for the ones that look like they are cut off.  There is a disease that affects seedlings that is called damping off.  It is a fungus that makes new seedlings rot at the ground level.  You can check the ones that look like they've fallen over - if it looks like stem just shriveled up it is damping off. It would be especially common when we have lots of rain.    For this you may want to start seeds indoors in sterile mix and transplant.  You can get seeds treated with a fungicide to reduce the likelihood of damping off, but you would probably have to order them from a catalog.