I(0) time series

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?

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.

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