Using a high-pressure synthesis method, we prepared the layered oxychalcogenide Ba2CoO2Ag2Te2 (space group: I4/mmm) with alternating stacks of CoO2 and Ag2Te2 layers, separated by Ba atoms. The CoO2 plane is greatly extended (Co-O = 2.19 angstrom on average) due to tensile strain from adjacent Ag2Te2 layers, causing displacement of oxide anions. Layered cobaltates with trans-CoO4X2 (X = chalcogen, halogen) coordination feature large spin-orbit coupling, which is linearly scaled by the tetrahedral factor of d(Co)(-X)/d(Co-O). However, applying this relation to Ba2CoO2Ag2Te2 yields a magnetic moment of similar to 4 mu(B), which is nearly twice the experimentally observed value of 1.87(17) mu(B). This result, along with a reduced Neel temperature (T-N = 60 K), originates from the off-centered position of otherwise under-bonded oxide anions, which changes the crystal field splitting of Co d orbitals.