The autocovariance of an Ornstein–Uhlenbeck process
$$ dX(t) = \theta (\mu - X(t))dt + \sigma dW(t) $$
is given on Wikipedia as
$$ Cov(X(s),X(t)) = \frac{\sigma^2}{2\theta}(e^{-\theta|t-s|} - e^{-\theta(t+s)}) \quad \quad (1).$$
which seems to depend on the time of origin since it has $t+s$ term.
On the other hand, the discreet-time analogue of the O-U process is the AR(1) process $$ X_{i+1} = \theta' (\mu' - X_i) + \sigma' Z_{i+1} $$
which has autocovariance (again according to Wikipedia)
$$Cov(X_{i+n},X_i) = \frac{(\sigma')^2}{1-(\theta')^2}(\theta')^{|n|} \quad \quad (2)$$
which only depend on the lag $n$. How does one reconcile the two? I can see that in the limit of $t,s \to \infty$ in such a way that $t-s = n$, $(1)$ becomes
$$ Cov(X(s),X(t)) = \frac{\sigma^2}{2\theta}e^{-\theta|n|} \quad \quad (3)$$
but it is not clear how this is related to $(2)$.
Also, if we have a time series of O-U realisation (for which we do not know the origin of time), what do we actually get when we compute sample autocovarince: $(1)$ or $(2)$?
Add 1
If I discretise the O-U process, then I get
$$ X_{t+1} - X_t = \theta (\mu - X_t) \delta t + \sigma \sqrt{\delta t} Z_{t+1} $$
or after re-arranging $$ X_{t+1} = \theta \mu \delta t + (1- \theta \delta t) X_t + \sigma \sqrt{\delta t} Z_{t+1} .$$
If I compare this now to $(2)$, I see that $\theta'= \theta \delta t - 1$ and $\sigma' = \sigma \sqrt{\delta t}$ so that on substitution into $(3)$ I get
$$ Cov(X(s),X(t)) = \frac{(\sigma')^2 /\delta t}{2(1+\theta')/\delta t}e^{-\theta|n|} = \frac{(\sigma')^2}{2(1+\theta')}e^{-\theta|n|}\quad \quad (4)$$
but it still does not have the form of $(2)$.