A closely related problem is encumbrance being also tied to movement, and running vs. walking distinction requires overhaul of reaction fire to work properly.
Are you mean something like a new button on the HUD where we could change the movement mode between run and walk?
In my eyes soliders always run currently due to I assume they want to reach the next cover asap, so I set the movement animation to maximum in settings screen which also save some of my real time.
If we want a distinction then I can suggest an alternative way where a run/walk button would be unnecessary. We already have a stun bar with regeneration, so if a runnig step (for 2TU) also consume some stun points and below a stun level (50%?) steps get +1TU penalty then it mean the solider can not run as fast as before, like slowed down to walk, where less stun points are consumed for each step. Stun level should not reduced further below a low value (30%?) to prevent sudden sleep just from walking.
I know this affect stunnig also but a tired person could fall into sleep more easily so this is logical and not a problem.
Reactionfire would be unaffected as far as I know.
Maybe this penalty could be applied in hard difficulties only. In the hardest difficulty even a smoother solution could be possible if +1TU/step is applied below 70% stun level (like jogging) and +2TU below 50% (walking).