Why the lag number of an AR model doesn't indicate the number of lags in a plotted ACF? In the below image, there are AR models with differing lags.  As far as I know, each autocorrelation function plot has an x-axis that is "number of lags".  Can someone help me understand how there are the same number of observations (lags, in the case of ACF) for each AR model?

 A: Your title asks about ACF but you actually display PACFs.
A lag-1 correlation induces a lag-2 correlation (and lag-3, 4 etc). Lag-1 and lag-2 correlations induce lag-3 and higher correlations, etc. So actual ACFs for AR models tend to show shrinking (and eventually, geometrically decreasing) correlations across lags.
e.g. for an AR(1), the population correlation at lag 1 is $\phi$, at lag 2 is $\phi^2$ and so on; at lag $k$ the correlation is $\phi^k$.
If you want a display that "cuts off" at the highest AR lag (apart from effects of noise) for a true AR model*, then you want the PACF (or the IACF).
In effect, the PACF removes the impact of the lower-order correlations on the higher order correlations, letting you see what remains after accounting for that.
If you look closely at your PACF plots in the question they do in fact cut off at the required lags. For some reason, the plots include the 0-lag - which is not informative - so the PACF for the AR(1) shows 2 spikes, one at lag 0 and one at lag 1 (and then cuts off to near 0), just as it should.
* in practice, real data don't exactly follow AR models so there's always a decision as to whether the model may be a close enough approximation to be useful for your purpose.
