B., I would STRONGLY caution you against the use of punishment, prohibition or ingnoring of the tantrums. I think your child is telling you that he has inner unhappiness which you need to find the root cause of. The crying/tantrumming episodes that you describe are his ONLY means of telling you he is anxious/insecure/sad/worried about something. You need to slow down, look at the situation a little more closely, find his "triggers (Your leaving his sight seems to be a big one)," then be really honest with yourself about what he's asking for. 2-year-olds are NOT manipulative, dishonest or independent - even though this is the underlying message in much of the popular advice. Toddlers want their parents' attention, unconditional acceptance (which means embracing EVERY one of their emotional expressions) and trust in their world - NOT to manipulate the world into their own pre-conceived idea of what it should be (as if they came out of the womb with a PLAN. Please). Toddlers are VERY easy to turn around with an empathy approach; but it takes consistency and extreme self-control.
Try this approach (if you don't already do this): When he's tantrumming next, take a deep breath, remember that he's a tiny person with an overwhelming emotion that he can't control, and that he is crying out for your help. Calmly tell him that you understand he's upset/disappointed/scared (etc.) and that you hear him and want to help him. Make sure he can't hurt himself or you, but stay close and DON'T ignore him (Ignoring him is the same as abandoning him: condtional love, based on BEHAVIOR). If you're in public, carry him off to the car. Sit with him until he comes out of it. He needs to know you're a support and that he is safe to express his feelings with you. Touch him and talk soothingly. When he's done, he should come to you for holding; if he doesn't initiate this, pick him up and hug him. Tell him, e.g.: "I know it's hard when you don't get ...(??), when Mommy has to go potty (etc.)." At this point, he'll most likely redirect himself, or lead you to engage with him in, an acceptable activity. You'll have to consistently do this for about a week before he starts to accept the change. Then he'll probably regress a little - swinging wildly the other way, to test your commitment.
Tantrums should not be embarrassing. People who think children should act like little robots are the ones with the "problem." Ignore the old ladies who glare at you for the way your child "misbehaves." By listening to your child's language and responding lovingly and empathetically, you are reinforcing the most IMPERATIVE foundation that your child will need throughout his life: his bond with his parents, and belief in their unconditional love. You can't shame him for FEELING. Children can be stifled and punished into "behaving" any way we want them to - but, as in everything, the easy way is usually the wrong way.
The other thought I had is that your part-time childcare may not be in sync with you, discipline-wise. Possibly, she is taking a more "cookie-cutter (1,2,3, timeout!)" approach - which is causing him to run to you for consoling and reassurance. I would double-check that.
I can't say it enough: "Smart Love" is the best parenting book! :)
Take care,
J. Smithson
Loving Hands Learn 'n' Play
http://www.lhlearnandplay.com
~edit: I just came back to read the additional responses and I have to thank Tanya for that WONDERFUL link! I have added it to my website and I'm ordering two of her books. I love that this philosophy is catching on. This is VERY encouraging! A flower for Tanya...:-)