Our knowledge about endohedral metallofullerenes (EMFs) is restricted to the structures with sufficient kinetic stability to be extracted from the arc-discharge soot and processed by chromatographic and structural techniques. For the most abundant rare-earth monometallofullerene MIII@C82, experimental studies repeatedly demonstrated C2v(9) and Cs(6) carbon cage isomers, while computations predicted equal stability of the "missing" C3v(8) isomer. Here we report that this isomer is indeed formed but has not been recovered from soot using standard protocols. Using a combination of redox extraction and subsequent benzylation and trifluoromethylation with single-crystal XRD analysis of CF3 adduct, we prove that Dy@C3v(8)-C82 is one of the most abundantly produced metallofullerenes, which was not identified in earlier studies because of the low kinetic stability. Further, using the Dy@C3v(8)-C82(CF3) and Dy@C3v(8)-C82(CH2Ph) monoadducts for the case study, we analyzed the role of metal-fullerene bonding on the single-ion magnetic anisotropy of Dy in EMFs. The multitechnique approach, combining ab initio calculations, EPR spectroscopy, and SQUID magnetometry, demonstrated that coordination of the Dy ion to the fullerene cage induces moderate, nonaxial, and very fluid magnetic anisotropy, which strongly varies with small alterations in the Dy-fullerene coordination geometry. As a result, Dy@C3v(8)-C82(CH2Ph) is a weak field-induced single-molecule magnet (SMM), whose signatures of magnetic relaxation are detectable only below 3 K. Our results demonstrate that metal-cage interactions should have a detrimental effect on the SMM performance of EMFs. At the same time, the strong variability of the magnetic anisotropy with metal position suggests tunability and offers strategies for future progress.