Unlike 遠雷, 熱界雷 doesn't exist as an ordinary Japanese word — real meteorological "heat lightning" is usually 熱雷 (nekkurai, thunder generated by summer-heat convection), without the extra 界 ("realm/boundary"). Gotouge's insertion of that extra kanji reads as a small but deliberate signal: this is the moment Kaigaku's blade work stops being pure Thunder Breathing and becomes fused with something else crossing a boundary into it — his Blood Demon Art heat, layered onto a technique he already knew as a human swordsman.
The translation problem runs the opposite direction from Form 3's "Thunder Swarm." There, a real Japanese idiom flattened into a generic English description; here, an invented Japanese compound becomes a perfectly ordinary, recognizable English weather term. "Heat Lightning" reads as completely natural to an English speaker, even though 熱界雷 itself is not a phrase any Japanese reader would recognize outside this manga.
Kaigaku spells out the mechanism himself in the very next line on the same page: an attack "with the sharpness of my blade, strengthened by Blood Demon Art," that "slashes and burns flesh." That's a demon's power stated outright, mid-duel, which is the clearest single piece of evidence in the whole sequence that this isn't a nostalgic flashback to two boys sparring — it's Kaigaku, already transformed, weaponizing forms Jigoro taught him as a human against the one classmate who still can't get past the First Form.