Instrumental variables - alternatives to 2SLS: not using exogenous variables in the first stage

I think that everyone knows how 2SLS works. say that we have:

$$y_t=a_1 +a_2X_t+a_3W^1_t+a_4W^2_t+e_t$$

Let's call this equation equation (1), where $$X_t$$ is an endogenous variable and $$W^1$$ and $$W^2$$ are exogenous variable.

A. in the first stage we regress endogenous variable on instrument $$Z$$ and all exogenous variable.

B. Then, in the second stage we replace $$X_t$$ with predicted values from the first stage($$\hat{X}$$), and estimate desired regression.

My question is the following:

In the (empirical) area of literature I am exploring (foreign exchnge intervention, where $$X_t$$ is intervention and $$y_t$$ is the exchange rate), what majority of authors have a theoretical model for $$X_t$$. What they do is then

A. regress $$X_t$$ on several theoretical predictors only, and estimate $$\hat{X}$$.This regression does not include $$W^1$$ and $$W^2$$

B. Because $$X_t$$ is endogenous, estimate (1) with $$\hat{X}$$ - a theoretically predicted $$X_t$$ - instead of $$X_t$$. Quite often, (1) is stimated with GARCH specifiation.

What kind of instrumental variable estimation is this, and is there any econometric justification for this approach? Is it an alternative to 2SLS (I see that some literature review papers name those papers as using 2SIV - two stage instrumental variables)? I was not able to find an answer to these question in the papers that use this methodology. If you want, I can provide more references