As safety is the top priority in mission-critical engineering applications, uncertainty quantification emerges as a linchpin to the successful deployment of AI models in these high-stakes domains. In this paper, we seamlessly encode a simple and principled uncertainty quantification module Spectral-normalized Neural Gaussian Process (SNGP) into GoogLeNet to detect various defects in steel wire ropes (SWRs) accurately and reliably. To this end, the developed methodology consists of three coherent steps. In the first step, raw Magnetic Flux Leakage (MFL) signals in waveform associated with normal and defective SWRs that are manifested in the number of broken wires are collected via a dedicated experimental setup. Next, the proposed approach utilizes Gramian Angular Field to represent the MFL signal in 1-D time series as 2-D images while preserving key spatial and temporal structures in the data. Thirdly, built atop the backbone of GoogLeNet, we systematically integrate SNGP by adding the spectral normalization (SN) layer to normalize the weights and replacing the output layers with a Gaussian process (GP) in the main network and auxiliary classifiers of GoogLeNet accordingly, where SN enables to preserve the distance in data transformation and GP makes the output layer of neural network distance aware when assigning uncertainty. Comprehensive comparisons with the state-of-the-art models highlight the advantages of the developed methodology in classifying SWR defects and identifying out-of-distribution (OOD) SWR instances. In addition, a thorough ablation study is performed to quantitatively illustrate the significant role played by SN and GP in the principledness of the estimated uncertainty towards detecting SWR instances with varying OODness.