When you're leased to a motor carrier, their policy covers you 'under dispatch' — hauling their loads, following their instructions. Non-trucking liability (NTL, aka bobtail) covers the truck when you're not.
Common misconception: NTL only responds when the truck is empty. Actually, it responds when you're not under dispatch — commuting home from the terminal, running personal errands, deadheading between contracts.
The claim gets ugly when 'under dispatch' is ambiguous. Both carriers point at each other. This is why NTL limits should match the underlying — $1M is standard.
NTL doesn't cover physical damage to your truck. That's a separate physical damage coverage you buy on your own policy — regardless of who you're leased to.
If you drop your lease and go independent, NTL doesn't turn into primary liability. You need to rewrite the policy with full primary coverage plus MCS-90 filing.