The terminology "fully Bayesian approach" is nothing but a way to indicate that one moves from a "partially" Bayesian approach to a "true" Bayesian approach, depending on the context. Or to distinguish a "pseudo-Bayesian" approach from a "strictly" Bayesian approach.
For example one author writes: "Unlike the majority of other authors interested who typically used an Empirical Bayes approach for RVM, we adopt a fully Bayesian approach" beacuse the empirical Bayes approach is a "pseudo-Bayesian" approach.
There are others pseudo-Bayesian approaches, such as the Bayesian-frequentist predictive distribution (a distribution whose quantiles match the bounds of the frequentist prediction intervals).
In this page several R packages for Bayesian inference are presented. The MCMCglmm is presented as a "fully Bayesian approach" because the user has to choose the prior distribution, contrary to the other packages.
Another possible meaning of "fully Bayesian" is when one performs a Bayesian inference derived from the Bayesian decision theory framework, that is, derived from a loss function, because Bayesian decision theory is a solid foundational framework for Bayesian inference.