I've worked with both those applications. MyChart is the portal you use (though I haven't really done much with the mobile app), and Care Everywhere is how the organizations talk to one another. I was involved in the very early work with Partners, getting Care Everywhere up and running, and also connecting to the state's HIE. Mt. Auburn has always seems to be slow at adopting the new functionality for patients...
A year ago I was at the ER at Mount Auburn and they did the basic blood work and then I waited for 2 hours and I finally asked a nurse what was going on and she said we are waiting for the results.
I handed her my phone and showed her the email I had gotten with the results an hour earlier