I can confirm that there is no agreement of what "extragradient method" really means. I know the interpretation by Christian Clason but I also know the one that is linked in Carlo Beenakkers answer. Let me add my point of view:

For simplicity consider a convex minimization problem
$$
\min_{x\in C} F(x)
$$
with a convex , lower semi-continuous function $F$ and a convex, closed set $C$.

Optima $u^*$ of this problem are characterized by
$$
u^* = P_C(u^* - \sigma\nabla F(u^*))
$$
where $P_C$ is the projection onto $C$ and $\sigma>0$. Doing a fixed point iteration for this equation gives
$$
u^{n+1} = P_C(u^n - \sigma\nabla F(u^n)),
$$
which is the standard projected gradient method.

Now we introduce a new variable $\bar u$ in the optimality condition and write it as
$$
u^* = P_c(u^* - \sigma\nabla F(\bar u)),\qquad u^* = \bar u.
$$
This seems artificial but now we can devise a new iteration and decide on different update rules for the artificial variable $\bar u$. The projected gradient method just takes $\bar u^{n+1} = u^{n+1}$. The *extragradient* method uses another *extra* gradient step for the new variable and reads as
$$
u^{n+1} = P_C(u^n - \sigma\nabla F(\bar u^n)),\qquad \bar u^{n+1} = P_C(u^{n+1} - \sigma \nabla F(\bar u^n))
$$
The benefit of this approach is that, although you make a new gradient step and usually have "better convergence", you have to evaluate the gradient of $F$ only *once* per iteration (but have to do two projections). If the cost for the evaluation of the gradient dominates the cost for the projection the, you may gain something.

A similar motivation for extragradient methods can be done for methods for variational inequalities, monotone inclusions or saddle-point problems…