The ARIANNA experiment is an Askaryan radio detector designed to measure high-energy neutrino induced cascades within the Antarctic ice. Ultra-high-energy neutrinos above $10^{16}$ eV have an extremely low flux, so experimental data captured at trigger level need to be classified correctly to retain as much neutrino signal as possible. We first describe two new physics-based neutrino selection methods, or "cuts", (the updown and dipole cut) that extend the previously published analysis to a specialized ARIANNA station with 8 antenna channels, which is double the number used in the prior analysis. The new cuts produce a neutrino efficiency of > 95% per station-year of operation, while rejecting 99.93 of the background (corresponding to 53 remaining experimental background events). When the new cuts are combined with a previously developed cut using neutrino waveform templates, all background is removed at no change of efficiency. In addition, the neutrino efficiency is extrapolated to 1,000 station-years of operation, obtaining 91%. This work then introduces a new selection method (the deep learning cut) to augment the identification of neutrino events by using deep learning methods and compares the efficiency to the physics-based analysis. The deep learning cut gives 99% signal efficiency per station-year of operation while rejecting 99.997% of the background (corresponding to 2 remaining experimental background events), which are subsequently removed by the waveform template cut at no significant change in efficiency. The results of the deep learning cut were verified using measured cosmic rays which shows that the simulations do not introduce artifacts with respect to experimental data. The data driven results in this paper demonstrate that the background rejection and signal efficiency of near surface antennas meets the requirements of a large scale future array, as considered in baseline design of the radio component of IceCube-Gen2.