I get the major heat and major itch with 2 things: May flies (which used to be in our area in the month of May, but their season has expanded a lot based on climate changes), and deer flies. May flies are black and much smaller than standard house flies, deer flies are the same size but with a patterned wing and not as black. There are also green head flies at a lot of beaches around here especially in the hot days of summer. The May fly redness can be 2-3 inches across especially if it's not right on the bone.
Since yours, per your SWH, has subsided into a center with surrounding redness, I'm guessing "bite" vs. itchy plant.
And I should say that I've added a third thing, as of this year - gypsy moth "stings" from the tiny hairs on their feet. First year ever, but we have a horrible infestation in New England so everyone is reacting. You can't see the little hairs - just the result of them. I realize a caterpillar is unlikely at the beach, but just in case you saw one later, or even sat on the ground where they are crawling, it's a distinct possibility. I sat down to do some planting, and got welts all over the back of my thighs and even my palms from when I used my hands to push up from a sitting position. However, the redness/itch is not instantaneous as with mosquitos or flies, but about 12 hours later.
I find rubbing anything on it at all is a bad idea - instead, I put ice on it, and then put an anti-itch cream on a bandaid and gently lay the bandaid across the center of the welt.
If you have a blister-type thing (which is what poison ivy turns into), try a compress of a salt-type remedy. When my son got horrible poison ivy, our pharmacist recommended Domeboro, which is a powder that's been around for decades. You mix it in water, then dip a washcloth in it, and let it sit on the blisters. No rubbing to aggravate the itching, and the blisters dry up faster.
Good luck!