This 2017 paper titled "Shape and Energy Consistent Pseudopotentials for Correlated Electron systems" defines energy-consistent psuedopotentials this way:
"A combined reproduction of core scattering, core polarisation, and
atomic excitation energies allows the generation of a new
pseudopotential from correlated electron calculations, referred to as
an energy consistent correlated electron pseudopotential (eCEPP)."
where by "reproduction", the abstract hints that they compare results using the pseudopotential to all-electron results. This means that they do calculations for core scattering, core polarization, and atomic excitation, all with every electron correlated, and then they do the same calculations but with the pseudopotential taking care of most of the electron correlation, and after comparing the two, if the psuedopotential method gives the same results as the all-electron calculation, then it is labelled as "energy-consistent".
Unfortunately that article does in my opinion, not as good of a job at defining what a shape-consistent pseudopotential is, but thanks to that, I had to look for a better resource to show you, and arrived at this 1999 paper ("On the accuracy of averaged relativistic shape-consistent
pseudopotentials") which defines both shape-consistent pseudopotentials and "energy-adjusted" pseudopotentials which gives some perspective on how the terminology can vary slightly from paper to paper. The "energy-adjusted pseudopotentials" or EAPP are described in the following way:
"The EAPP are based on quantum mechanical observables (atomic
excitations, ionization energies, etc. and the starting point is
either a quasi-relativistic or a 4-component all-electron
calculation."
So again, they are comparing energies from relativistic all-electron calculations to energies obtained with the pseudopotential, but the specific energies that are being compared are not necessarily exactly the same from paper to paper.
They also describe shape-consistent pseudopotentials or SCPPs, and as suspected in my initial comment, this has to do with the accuracy of the shape of the wavefunction:
"In contrast to the previous method, the SCPP are based on orbital properties"
I'll also mention that this answer by Susi Lehtola points out that energy-consistent PPs are shape-consistent, but shape-consistent PPs are not necessarily energy-consistent, and refers the reader to this excellent review paper: "The Pseudopotential Approximation in Electronic Structure Theory." That paper describes shape-consistent potentials in the following way:
"If the pseudopotential parameters are fitted in such a way that the
valence orbitals of different symmetries reproduce to a high accuracy
the corresponding all-electron orbitals from a certain cut-off
(valence) radius $r_c$ onwards with matching orbital energies, we obtain
shape-consistent pseudopotentials."
