I think you're already using the most recent text and familiar with the rulings involved here, but because there was an errata that pretty seriously altered how the system treats Unarmed Strikes, I'll reproduce the the latest version here:
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.
And the Sage Advice Compendium contains an important note relevant to this discussion:
What does “melee weapon attack” mean: a melee attack with a weapon or an attack with a melee weapon?
It means a melee attack with a weapon. Similarly, “ranged weapon attack” means a ranged attack with a weapon. Some attacks count as a melee or ranged weapon attack even if a weapon isn’t involved, as specified in the text of those attacks. For example, an unarmed strike counts as a melee weapon attack, even though the attacker’s body isn’t considered a weapon.
Here’s a bit of wording minutia: we would write “melee-weapon attack” (with a hyphen) if we meant an attack with a melee weapon.
So, to your question:
The distinction between "weapon attack" and "attack with a weapon" (or "attack using a weapon" in this case, which is completely synonymous) is a rather fine distinction of language that isn't always used logically -- especially within the Player's Handbook itself, where the errata'd rules for Unarmed Strike had some potentially unintended side-effects.
For example, technically speaking, enlarge/reduce does not change the damage dealt by unarmed strikes, which makes no logical sense, and according to Sage Advice, a half-orc's Savage Attacker doesn't make your unarmed crits hit harder because your fists don't have 'weapon damage dice' even when you have an ability that makes your unarmed strike deal damage dice.
So by the rules as written in the errata, you can't use the Fire Rune with unarmed strikes: hitting with your fists is specifically not hitting with a melee weapon, though it is a "melee weapon attack".
However, I'm not sure that was actually intentional. While you do have to inscribe the rune on an object, it doesn't have to be inscribed on a weapon in order to use the flaming shackles effect, and there seems to be no balance or flavor reason that you'd need to be hitting somebody with an actual weapon to activate a rune on a shield or headband or something. If your DM wants to allow you to use a Fire Rune with your fists, there probably is no real harm to it. A lot of rules about what can or can't be done without a weapon are more for flavor than balance, and altering the flavor of you class abilities is something you should always feel free to discuss with the person running your game.