What's this reaction called after laser hair removal on legs for the first time? (Photos)

From your photos and your history of improvement with cortisone cream, my assessment would be that you have a follicular reaction.  Sometimes when the root of the hair treated with laser is killed but the shaft remains in the pore, the body will develop an allergic-like reaction to the remaining hair which results in redness, localized swelling and itchiness.  The reaction is usually transient and resolves once the hair shafts fall out.  Topical cortisone creams or oral Benedryl will help.  As always, it is best for your treating physician or clinic to see you in person to confirm the diagnosis.  If they confirm this assessment, taking Benedryl before the next procedure can make it less likely that this might happen again.

Thanks for your question and posting pictures.  It looks like you might have had folliculitis.  Your skin should be getting back to normal soon.  Sometimes this happens on its own, but not normally.  Don't let this dissuade you from doing additional treatments.  After LHR and most other laser treatments, it is important to not raise your body temperature for the next several hours.  Avoid strenuous exercise, hot tubs or even a hot bath for the rest of the day.  Aloe Vera my help soothe your skin.  I hope this helps.