Is detrending a variable equivalent to including a time trend in the regression? Suppose that I have determined that one or more of the variables in my linear model is trend stationary. Does it suffice to then simply include the trend terms in my model, or do I have to detrend (by explicity subtrating out the trend)?
My hand-wavy argument is as follows. Suppose that I wish to regress $y_t$ on $x_t$ but discover that $y_t$ is linear-trend stationary ($x_t$ is already stationary, say). If I fit the model
$$
 y_t = \beta_0 + \beta_1 x_t + \beta_2 t + \epsilon_t\ ,
$$
I can rearrange this to obtain
$$
 y_t - \beta_0 - \beta_2 t = \beta_1 x_t  + \epsilon_t\ ;
$$
is it true that the LHS "equals" (this is the hand-wavy part) the residuals $\delta_t$ of the model
$$
 y_t = \gamma_0 + \gamma_1 t + \delta_t\ ?
$$
If this were to be the case, I would think that one could detrend $y_t$ by simply including time terms of the desired order.
Does this argument work? (The upside would be easy interpretability of my regression coefficients. The model would also be easier to work with in a VAR context etc.)
If my argument does not work, similarly to this question then, I would like to understand how one might go about interpreting the coefficients of a linear regression of $\hat{y_t}$ on $\hat{x_t}$ when I have detrended one or both of the variables. (It must be said here that one possible benefit of this method is when my sample is extremely small; detrending before fitting allows me to save a crucial few df's.)

As @whuber points out in the comments, this question is in fact similar to an existing post. For anyone interested in reference that explicitly deals with time trends, the argument can be found in Woolridge's, "Introductory Econometrics", Ch. 10, in the section entitled "A detrending interpretation of regressions with a time trend". The argument is the same "partialling out interpretation of OLS" that @whuber discusses in the above post.
In sum, the answer to the posted question is yes: including a time trend (or trends, of desired order) is equivalent to detrending. More precisely, the estimate $\hat{\beta_1}$ obtained by regressing $y_t$ on $x_t$ is exactly the same as that obtained by regressing $\dot{y_t}$ on $\dot{x_t}$, where $\dot{y_t}$ and $\dot{x_t}$ are the residuals obtained by regressing $y_t$ and $x_t$ on $t$ respectively.
