The bounties available today seemed perfect for testing this out, so I picked up 2 bounties from Shaw Han and one from Zavala:
BREAD AND BUTTER
Basic / Cosmodrome Bounty
Defeat combatants in the Cosmodrome using any Kinetic weapon.
Kinetic weapons do not have an element associated with them.
SPECIALITIES
Basic / Cosmodrome Bounty
Defeat combatants in the Cosmodrome with any Energy weapon.
Energy weapons have an element associated with them. Hitting combatant
shields that match the weapon's element will deal bonus damage and
cause an elemental explosion.
COLD CALCULATIONS
Basic / Daily Bounty
In strikes, defeat opponents with Stasis damage and defeat
opponents that are slowed or frozen by Stasis.
I then loaded into the Devil's Lair strike in the Cosmodrome and tried my absolute hardest to progress the energy weapon kills bounty with Cryosthesia.
A normal kill with the weapon progressed the kinetic weapon kills bounty and the stasis bounty, but not the energy weapon bounty. Not sure how to embed video, so here is a link to a clip of the bounty progression after the first such kill.
I froze the next target, then killed it with Cryosthesia (even though a Hive Knight really really didn't want me to). This progressed my kinetic weapon kills bounty by +1, and the stasis kill bounty by +3 (possibly due to stacking the events "stasis kill" with the weapon and a "combatant frozen with stasis" kill, since the bounty has multiple different conditions for progression and I don't know how they're weighted). The energy weapon bounty never progressed.
It doesn't seem possible to kill anything with just the Cryosthesia freeze, since neither the stasis freeze nor the opponent breaking free deals any damage at all, so I believe normal kill or freeze-then-normal-kill are the only options here, and neither count as an energy weapon kill. Both options count as both a kinetic weapon kill and a stasis kill.