I recently came across the Dickey-Fuller test for existence of a unit root in an AR(1) series, definition on Wikipedia. If a unit root exists, the series is not stationary. Fine by me.
Now looking at some applications and interpretations of the Dickey-Fuller test, apparently people say that if the null hypothesis is rejected, there is evidence that the process is stationary. More so, this "logic" is apparently still applied in case time series are obviously seasonal or other time-dependent things go on. I realise that there also is an augmented Dickey-Fuller test that allows to detect unit roots for some more sophisticated models, but anyway...
The thing that bothers me is the following. Stationarity is a standard model assumption in time series analysis. It's quite restrictive in my view, any time-dependent pattern is not allowed. Normally when testing model assumptions (e.g., normality, independence...), the restrictive model assumption is the null hypothesis and the data can reject it or not, but we will never have evidence in favour of the model assumption, as this is an idealisation, will not hold precisely, and we can be happy enough if it's just not obviously incompatible with the data.
For the Dickey-Fuller test it's apparently the opposite. Stationarity is the alternative, rejecting the unit root amounts to rejecting non-stationarity, or, in other words, to observe more or less strong evidence for stationarity. This seems to be a misinterpretation to me, because there are lots and lots of possibilities to have non-stationary series that do not fulfill the Dickey-Fuller unit root model (seasonal series to start with), and may therefore lead to rejection of the unit root model. So this doesn't seem to provide positive evidence in favour of stationarity at all; the only thing is that one specific form of non-stationarity is ruled out.
Am I misunderstanding something, or is it indeed the case that rejection of a unit root is pervasively misinterpreted?