How Does Storm's Gift interact with Harold of Thunder?

These are Storm's Gift's stats according to the wiki:
(20-30)% increased Damage over Time
(250-300)% increased Evasion and Energy Shield
+(20-30)% to Lightning Resistance
Enemies you kill are Shocked
Shocks you inflict spread to other Enemies within a Radius of 15

These are Harold's stats according to the wiki:
Buff grants (2-19) to (7-76) Added Spell Lightning Damage
Buff grants (2-19) to (7-76) Added Attack Lightning Damage
Deals (1-23) to (31-1108) Lightning Damage
Base duration is 6 seconds
Storm Hits an Enemy every 0.25 Seconds
Cannot inflict Shock
Creates a storm when you Kill a Shocked Enemy

In general, would the gloves make it so killing any enemy with anything count as killing a shocked enemy(since the enemies you kill are shocked) and thus trigger Harold's zappy gun?

Would the gloves still cause the AoE shocking effect if an enemy was killed by Harold's zappy gun?

this 3.15 build I've found makes use of them, but I don't see it clearly stating there is an interaction between Gift and HoT.

All I can speculate is that HoT needs for the enemy to be shocked before it gets killed, and not on the killing blow. Perhaps a Necromancer user can help.