I suspect there is no general term that will cover all cases. Consider, for example, a white noise generator. In that case, we would just call it white noise. Now if the white noise comes from a natural source, e.g., AM radio band white noise, then it has effects including superimposed diurnal, seasonal, and sun-spot (11 year) solar variability, and man made primary and beat interference from radio broadcasts.
For example, the graph in the link mentioned by the OP looks like amplitude modulated white noise, almost like an earthquake. I personally would examine such a curve in the frequency and or phase domain, and describe it as an evolution of such in time because it would reveal a lot more about the signal structure by direct observation of how the amplitudes over a set of ranges of frequencies evolve in time with respect to detection limits as opposed to thinking about stationarity, mainly by reason of conceptual compactness. I understand the appeal of statistical testing. However, it would take umpteen tests and oodles of different criteria, as in the link, to incompletely describe an evolving frequency domain concept making the attempt at developing the concept of stationarity as a fundamental property seem rather confining. How does one go from that to Bode plotting, and phase plotting?
Having said that much, signal processing becomes more complicated when a "primary" violation of stationarity occurs; patient dies, signal stops, random walk continues, and so forth. Such processes are easier to describe as a non-stationarity than variously as an infinite sum of odd harmonics, or a decreasing to zero frequency. The OP complaint about not having much literature to document secondary stationarity is entirely reasonable; there does not seem to be complete agreement as to what even constitutes ordinary stationarity. For example, NIST claims that "A stationary process has the property that the mean, variance and autocorrelation structure do not change over time." Others on this site claim that "Autocorrelation doesn't cause non-stationarity," or using mixture distributions of RV's that "This process is clearly not stationary, but the autocorrelation is zero for all lags since the variables are independent." This is problematic because auto-non-correlation is typically "tacked-on" as an additional criterion of non-stationarity without much consideration given to how necessary and sufficient that is for defining a process. My advice on this would be first observe a process, and then to describe it, and to use phrases crouched in modifiers such as, "stationary/non-stationarity with respect to" as the alternative is to confuse many readers as to what is meant.
Here's a simple example illustrating why the answer is no.
Let $$P = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix}$$ be the transition matrix for a first-order Markov process $X_t$ with state space $\left\{0, 1\right\}$. The limiting distribution is $\pi = \left(0.5, 0.5\right)$. However, suppose you start the process at time zero with initial distribution $\mu = \left(1, 0\right)$, i.e. $X_0 = 0$ with probability one.
We then have $\mathbf{E}[X_0] = 0 \neq \mathbf{E}[X_1] = \frac{1}{2}$, meaning the moments of $X_t$ depend on $t$, which violates the definition of stationarity.
Here's some R code illustrating a similar example with
$$P = \begin{pmatrix} 0.98 & 0.02 \\ 0.02 & 0.98 \end{pmatrix}$$
p_stay <- 0.98
P <- matrix(1 - p_stay, 2, 2)
diag(P) <- p_stay
stopifnot(all(rowSums(P) == rep(1, nrow(P))))
mu <- c(1, 0)
pi <- matrix(0, 100, 2)
pi[1, ] <- mu
for(time in seq(2, nrow(pi))) {
pi[time, ] <- pi[time - 1, ] %*% P
}
plot(seq(1, nrow(pi)), pi[, 1], type="l", xlab="time", ylab="Pr[X_t = 0]")
abline(h=0.5, lty=2)
The fact that $X_t$ is converging in distribution to some limit does not mean the process is stationary.
Best Answer
Stationarity is a property of a stochastic process. A perfect sine wave is not a stochastic process. Hence, it can't be stationary or non-stationary. It doesn't have any random parts.
$$y_t=\sin (\phi t+\theta)$$
It's like asking whether a song is black or white. The music has no color, it has many other properties but color is not one of them.
Now, you could look at the problem differently. As you wrote the phase and frequency are unknown. So, if you look at the family of processes: $$y_t=\sin (\phi_i t+\theta_i)$$ Where $\phi_i,\theta_i$ come from some distribution, and you're to estimate $E[y_t]$, then it's a more interesting question. It's still not a stochastic process though.
The stochastic process represents an evolution of random variables. In the case of a perfect sine wave it's entirely defined by two random variables $\phi_i,\theta_i$. There's no evolution.
In other words there's got to be some kind of randomness and uncertainty introduced as time progresses in order for the process to be stochastic. In your case all the uncertainty is introduced at time 0.