I see most of this through the lens of representation. Obama and Congressional Democrats made promises in 2008; they interpreted them; and those interpretations guided what they tried to do in office. “Promises” here means not just specific public policy choices, but also governing style, priorities, and all the rest of behavior in office. Different campaign in 2008, different promises, different interpretation, and therefore different behavior in office.
Getting back to Ezra Klein (who emphasized the limits on presidential-led change, especially from "going public") and Andrew Sprung (who emphasized the way the 2008 election did really produce results): the answer, then, is that they’re both right. No, Obama couldn't go over the heads of Congress in summer 2009 to force action on health care, or over the heads of Congress later to force action on his jobs bill. Any presidential action of that sort is at best going to produce extremely limited success in normal times, and can easily backfire.
But it's also true that elected officials in democracies generally try to keep their promises; they try to have healthy representative relationships with their constituents, which really does seem to involve taking seriously the job of making promises, interpreting them, and acting while in office with that interpretation in mind. It doesn't mean that there's an absolute one-to-one correspondence between everything a politician ever says and what they do; that's not how it seems to work. It's more that they try to be who they say they are, and that creates a situation in which they do, or attempt to do, more or less what they said they would do. Or at least some version of it.
Thinking this through turns out to be extremely difficult...we know both that politicians really do seem to take their promises seriously, but also that most actual, real-world constituents don't pay very much attention to this stuff at all, and that even for those who do it's hard to see how the interpretations of the constituency are related to the interpretations of the politician. I do have an answer to it; it has to do with party politics as the practical real-world link between politicians and constituents (well, that's some of it; it also has something to do with Act 4 of Henry V, the "little touch of Harry", but I'll save all the explanations for some other time). The shortish answer is that, yup, campaigning and electioneering -- the "outside" part of it -- matters a whole lot.
Hey, it's Friday late afternoon; I'm sure I'll get back to this some other time. As you can see, I have a lot of thoughts on this one, even if this post is a bit scattered. Sorry about that. The quick answer to how Klein and Sprung are both right, at any rate, is: representation.