Physically understanding autoregressive model (really basic question)
Your understanding is right. Just a few points...


I think you meant Yule-Walker equations.
It is better to distinguish $N$ from $n$. In an AR($N$) model we consider last $N$ terms to approximate the next term. 
An important interpretation that you missed was the fact that when we consider
$$\epsilon=x[n]-\tilde{x}[n]=x[n]-\sum_ka_kx[n-k],$$ or in a more standard way (representing the process by variable $y$ instead of $x$
$$y[n]=\sum_ka_ky[n-k]+\epsilon[n],$$
it can be seen as an input-output relationship of an LTI filter, $\epsilon[n]$ being input and $y[n]$ the output. More specifically, 



   AR model can be seen as the output of an all-pole IIR filter which is excited by a white noise input.