To go even further, this should be a mandated policy city by city. And it should be a blanket policy that applies to any hot beverage being sold. Just like a shopping bag.
1. All shops serving hot beverages require a mug. The mug may be purchased at any shop covered by the policy, or otherwise bring your own. Warning about 'Caution, Hot Beverage inside!' become moot. Your cup , your problem.
2. Purchased mug is sold at small profit and first fill-up is free or whatever incentive to purchase the liquid additionally with the mug.
3. Purchases that use a refillable mug, get a 5% discount during phase-in.
4. Shops do not offer to charge extra for the disposable cups. If available, this lazy option would give a choice that leaves them justified in being lazy, and creates a sort of contract that they still contribute to the greater good.
5. Add a creative incentive that allows purchasers to add value to this activity. Marketers can add to this.
6. Make all these mugs as easy to wash and as durable as possible. Non-toxic, non-breakable, easy to wash. No problem.
7. People love to drink coffee/tea in their cars. There's an another possible angle to create a cross-incentive
Well there's a few things I thought about.