Definition of I(0) and its relation to stationarity I thought that I(0) time series were simply stationary time series. But the definition on the Wikipedia page says:

A time series is integrated of order 0 if it admits a moving average
  representation with: $$ \sum_{k=0}^\infty |b_k|^2 < \infty $$
  Therefore, all stationary processes are I(0), but not all I(0)
  processes are stationary.

Is the definition on the page correct?
 A: It's correct because of the subtle difference between stationarity and wide-sense stationarity (WSS). All $I(0)$ are wide-sense (week-sense) stationary, or in other terms, covariance stationary, as implied in the first sentence of the wiki entry of Order of Integration. This doesn't mean that the series is stationary, (i.e. strongly stationary). But, a stationary series is by definition WSS. The WSS requires mean to be constant, and covariance to depend on only the time difference between the samples. But, the stationarity is a far stronger requirement. You might see that the term stationarity is being used in some sources to refer to WSS for simplicity. 
A: The concept of order of integration is not uniquely defined, and there exist multiple definitions that to a large degree overlap. Some of the more hand-wavy definitions rather imprecisely define it as "the number of differences needed to achieve stationarity". Engle and Granger (1987) and Johansen (1995) define the concept more precisely:

A series with no deterministic component which has a stationary,
  invertible ARMA representation after differencing $d$ times is said to
  be integrated of order $d$ (Engle and Granger, 1987)
A stochastic process $Y_t$ which satisfies
  $Y_t-E(Y_t)=\sum_{i=0}^\infty C_i\epsilon_{t-i}$ is called $I(0)$ if
  $\sum_{i=0}^\infty C_iz^i$ converges for $|z_i|<1$ and
  $\sum_{i=0}^\infty C_i\neq 0$. (Johansen, 1995)

One important implication stemming from these definitions is that an MA(1) process with $\theta=1$ is not $I(1)$ (because it is not invertible, and because $\sum_{i=0}^\infty C_i=1-1=0$). So according to these definitions, not all weakly stationary processes are stationary.
The definition on Wikipedia is therefore neither correct nor incorrect; it is merely one definition. For more discussion, see Davidson (2009).

Davidson, James (2009) "When is a time series I(0)?". Chapter 13 of The Methodology and Practice of Econometrics, a festschrift for David F. Hendry edited by Jennifer Castle and Neil Shepherd, Oxford University Press. https://people.exeter.ac.uk/jehd201/WhenisI0.pdf
Engle, Robert & Granger, Clive (1987) Co-Integration and Error Correction: Representation, Estimation, and Testing. Econometrica, 55(2), 251-276. doi:10.2307/1913236
Johansen, Soren (1995) "Likelihood-Based Inference in Cointegrated Vector Autoregressive Models", Oxford University Press. https://www.oxfordscholarship.com/view/10.1093/0198774508.001.0001/acprof-9780198774501
