They are definitely tuning into one another's message. Cameron is clearly worried and Obama is his trump card. But it looks terrible and Obama has damaged himself with many in the UK by appearing to say - stay in the EU or you go to the back of the queue with the US. Some special relationship!
It is kinda obvious that the UK Obama wants is subservient to US interests. This was exposed during his intervention, flatteries aside.
I agree it is in the US interest for Britain to remain in the EU. It isn't US sovereignty at stake. But from a British perspective, we are being dragged into a European superstate and it is our interest to decide if we want that or not. I don't. Obama looks hypocritical in pushing something for the UK which would be entirely unacceptable to the US people. Americans would never accept participation in an American political block controlled by others. Damn the convenience to third parties in negotiating a single trade deal. The world just happens to be a little bit complicated and we all live with that.
He should have avoided partisan engagement. Instead, I suspect he will have improved the prospects for those who wish to leave, as standing alone is a powerful narrative in UK history.
In my view, there is a relationship with the EU which could work for the EU and the UK. But this isn't it.
By the way, the EU is far from a British construction. It is largely a French and German enterprise. Britain has more ancient ties with other Commonwealth countries and in my view it should refocus on those.
I don't think Obama is Cameron's parrot. He was expressing his views. Indeed, it looks to me as if Cameron's position is as Obama's footstool. Cameron just says what the US President requires.