"De-meaning" or "Differencing the mean of..." in mathematical term In a standard regression literature, the following terms are used almost interchangeably and are used also loosely:

*

*"De-meaning the equation gives..."

*"Differencing the mean of the outcome eliminates...."

*"The mean-difference provides..."

Is there a rigorous and unequivocal way to define these terms using the conditional expectation from probability theory?
 A: It is hard to comment without context. Many terms may be ambiguous or there may be different procedures and methodologies of doing things that at first sight may be the same, but because of the "technical details" are not. When reading terms like this, they should always be accompanied by the definitions of the terms and the actual methodology that was used. If they are not, it is a guessing game.
Referring to the quotes, the usual meaning of the three phrases would be different.
1. "De-meaning the equation gives..."
De-meaning usually means subtracting the mean from all the values. If the mean is
$$
\bar x = \frac{1}{N} \sum_{i=1}^N x_i
$$
then de-meaning is the operation that produces $x_i' = x_i - \bar x$ for all observations $i=1 \dots N$.
2. "Differencing the mean of the outcome eliminates...."
This one might mean different things. For example, similar language was used here to describe the difference-in-differences method:

The DID strategy relies on two differences. The first is a difference
across time periods. Separately for the treatment group and the
control group, we compute the difference of the outcome mean before
and after the treatment. This across-time difference eliminates
time-invariant unobserved group characteristics that confound the
effect of the treatment on the treated group. But eliminating
group-invariant unobserved characteristics is not enough to identify
an effect.
[...] The ATET is then consistently estimated by
differencing the mean outcome for the treatment and control groups
over time to eliminate time-invariant unobserved characteristics and
also differencing the mean outcome of these groups to eliminate
time-varying unobserved effects common to both groups. [...]

Here it is about the difference between means.
3. "The mean-difference provides..."
"Mean-difference" would usually mean that you calculate the mean of differences. For example, you have the $z_i = x_i - y_i$ datapoints for $i = 1 \dots N$ and calculate mean-difference, i.e. the mean of $z_i$'s.
So the terms are not interchangeable.
