In neuromorphic computing, artificial synapses provide a multi-weight conductance state that is set based on inputs from neurons, analogous to the brain. Additional properties of the synapse beyond multiple weights can be needed, and can depend on the application, requiring the need for generating different synapse behaviors from the same materials. Here, we measure artificial synapses based on magnetic materials that use a magnetic tunnel junction and a magnetic domain wall. By fabricating lithographic notches in a domain wall track underneath a single magnetic tunnel junction, we achieve 4-5 stable resistance states that can be repeatably controlled electrically using spin orbit torque. We analyze the effect of geometry on the synapse behavior, showing that a trapezoidal device has asymmetric weight updates with high controllability, while a straight device has higher stochasticity, but with stable resistance levels. The device data is input into neuromorphic computing simulators to show the usefulness of application-specific synaptic functions. Implementing an artificial neural network applied on streamed Fashion-MNIST data, we show that the trapezoidal magnetic synapse can be used as a metaplastic function for efficient online learning. Implementing a convolutional neural network for CIFAR-100 image recognition, we show that the straight magnetic synapse achieves near-ideal inference accuracy, due to the stability of its resistance levels. This work shows multi-weight magnetic synapses are a feasible technology for neuromorphic computing and provides design guidelines for emerging artificial synapse technologies.