Those spots remind be of something -- I know I've seen them before -- but I can't pin point what it is. It does look like splash backs from the soil caused it -- have you been spraying the seedlings? I don't think it's insect damage.
I would try spraying with 10% milk solution (1 part milk:10 part dechlorinated water) in case it's fungal disease. If you have compost tea, that might help too. If you have AACT (sticky on top of Organic Gardening Forum), that would be even better. I would water with the compost tea as well as spray.
It could be sucking insect damage, but I'm with AS; more likely disease.
A sticky card to check for insects might be a good idea to eliminate that thought entirely, and a little soil topdress with good compost (just enough to cover the soil), should help with splash-up diseases (pretty common).
So I am indeed spraying them with a mister. If it is splash-back, would you still use that milk mixture? Does it matter if it is whole 2% or skim milk?
Will this go away and more importantly will it spread to other plants (brassicas?)
I had something that looks almost exactly like that on my 4" tomato seedlings after spraying them with compost tea in the late afternoon, which then went in a mini greenhouse. Oops.
I set them out in the sun and sprayed with a diluted whole milk and corn starch mix. The spots did not progress much, but the affected leaves eventually yellowed and dried, so I cut them off. Was it fungal? bacterial? leaf edema? I'm not sure, but its gone (for now at least).
Could have been sunburn (water drops magnifying hot sun and burning leaves). Could have been any of the others you listed. SOunds like you turned the corner...?