We report the results of neutron elastic-scattering measurements made between -250 degrees C and 620 degrees C on the lead-free relaxor (Na1/2Bi1/2)TiO3 (NBT). Strong, anisotropic, elastic diffuse scattering intensity decorates the (100), (110), (111), (200), (210), and (220) Bragg peaks at room temperature. The wave-vector dependence of this diffuse scattering is compared to that in the lead-based relaxor Pb(Mg1/3Nb2/3)O-3 (PMN) to determine if any features might be common to relaxors. Prominent ridges in the elastic diffuse scattering intensity contours that extend along < 110 > are seen that exhibit the same zone dependence as those observed in PMN and other lead-based relaxors. These ridges disappear gradually on heating above the cubic-to-tetragonal phase transition temperature T-CT = 523 degrees C, which is also near the temperature at which the dielectric permittivity begins to deviate from Curie-Weiss behavior. We thus identify the < 110 >-oriented ridges as a relaxor-specific property. The diffuse scattering contours also display narrower ridges oriented along < 100 > that are consistent with the x-ray results of Kreisel et al. [Phys. Rev. B 68, 014113 (2003)]; these vanish near 320 degrees C, indicating that they have a different physical origin. The < 100 >-oriented ridges are not observed in PMN. We observe no equivalent relaxor-specific elastic diffuse scattering from the homovalent relaxor analogues K0.95Li0.05TiO3 (A-site disordered) and KTa0.95Nb0.05O3 (B-site disordered). This suggests that the < 110 >-oriented diffuse scattering ridges are correlated with the presence of strong random electric fields and invites a reassessment of what defines the relaxor phase. We find that doping NBT with 5.6% BaTiO3, a composition close to the morphotropic phase boundary with enhanced piezoelectric properties, increases the room-temperature correlation length along [1 (1) over bar0] from 40 to 60 angstrom while doubling the associated integrated diffuse scattering. Similar behavior was reported by Matsuura et al. [Phys. Rev. B 74, 144107 (2006)] for morphotropic compositions of PMN doped with PbTiO3. Finally, we comment on the recent observation of monoclinicity in NBT at room temperature by placing a strict bound on the strength of the (1/2 1/2 1/2) superlattice reflection associated with the Cc space group based on the atomic coordinates published in the x-ray study by Aksel et al. [Appl. Phys. Lett. 98, 152901 (2011)] for NBT. We show that a skin effect, analogous to that reported in the relaxors PZN-xPT and PMN-xPT, can reconcile our neutron single-crystal data with the x-ray powder data of Aksel et al. [Appl. Phys. Lett. 98, 152901 (2011)]. Our finding of a skin effect in a lead-free, A-site disordered, heterovalent relaxor supports the idea that it arises in the presence of strong random electric fields.