Difference Between Two Forms of Equations of Auto Regressive (AR) Model
Auto Regressive Model means the current output is a linear combination of previous outputs and driving noise:

$$ y \left[ n \right] = \sum_{k = 1}^{p} {a}_{k} y \left[ n - k \right] + v \left[ k \right] $$

As you can see, the current value $ y \left[ n \right] $ depends on $ m $ values before it.
The parameter $ p $ is the order of the model while $ v \left[ k \right] $ is the driving IID White Noise.

Usually if we want to estimate the parameters of the model we have 2 main cases:


The Model Order $ p $ Is Known
In this case we need to estimate $ p $ parameters $ \left\{ {a}_{k} \right\}_{k = 1}^{p} $ from a set of $ m \geq p $ samples $ \left\{ y \left[ n \right] \right\}_{n = 1}^{m} $. the classic way to do so is using the Yule Walker Normal Equations.
The Model Order $ p $ Is Unknown
If $ p $ is unknown the intuitive a approach is estimate it and then go back to (1). One could estimate is by running on a grid of options (Try model order 3, 4, 5, 6, ... and chose the one which works best in MSE) yet this might do overfit.  A classic approach to evaluate the grid is using Akaike iInformation Criterion (AIC).


In your equations:



The connection is between the output to a different signal which means it is not an AR model.

If you tell us where did you get those form we might be able to clear things up.

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