In a MA(1):
$$ y_t = \mu + \epsilon_t + \theta \epsilon_{t-1} $$
What is the error term $ \epsilon $? So far I've read that it can be:
Difference between data and forecasted value
Difference between previous values: $ y_t - y_{t-1} $
Variation of other time series, like variation of temperature is $\epsilon$ and lemonade demand is $y_t$
Are all of those interpretations OK? If so what is the underlying property tying them together?
The first one is incorrect unless you know the true conditional expectation of the random variable being forecast, $\mathbb{E}(y_t|I_{t-1})=\mu+\theta\varepsilon_{t-1}$, and use it as a forecast. This is unlikely in practice as you do not know $\mu,\theta,\varepsilon_{t-1}$, you may only have imperfect estimates of each of them.
The second one is incorrect. You can easily check that by expressing $y_t$ and $y_{t-1}$ in terms of $\varepsilon$s: \begin{aligned} y_t-y_{t-1} &= (\mu+\varepsilon_t+\theta\varepsilon_{t-1})-(\mu+\varepsilon_{t-1}+\theta\varepsilon_{t-2}) \\ &=\varepsilon_t-(1-\theta)\varepsilon_{t-1}-\theta\varepsilon_{t-2} \\ &\not\equiv\varepsilon_t. \end{aligned}
The third one is possibly correct if we replace variation with value. E.g. suppose you are producing some good, stockpiling some of it and selling the rest. Your production in time period $t$ is $x_t=m+\varepsilon_t$ with $\mathbb{E}(\varepsilon_t)=0$ and your stock is $y_t$. The sequence of $\varepsilon$s is i.i.d. A $1-\theta$ fraction of the period's production is sold during the next period, and the remaining $\theta$ during the one after that. Then your stockpile is \begin{aligned} y_t&=x_t+\theta_1x_{t-1} \\ &=\mu+\varepsilon_t+\theta_1\varepsilon_{t-1}, \end{aligned} where $\mu=(1+\theta_1)m$. Thus, $y_t$ follows and MA(1) process.
For real-life examples of moving-average processes, see e.g. this, this and this as well as other answers in that thread. (The example to illustrate the third one above is borrowed from the latter answer.)