Lipid phase heterogeneity in the plasma membrane is thought to be crucial for many aspects of cell signaling, but the physical basis of participating membrane domains such as "lipid rafts" remains controversial. Here we consider a lattice model yielding a phase diagram that includes several states proposed to be relevant for the cell membrane, including microemulsion-which can be related to membrane curvature-and Ising critical behavior. Using a neural-network-based machine learning approach, we compute the full phase diagram of this lattice model. We analyze selected regions of this phase diagram in the context of a signaling initiation event in mast cells: recruitment of the membrane-anchored tyrosine kinase Lyn to a cluster of transmembrane IgE-FcεRI receptors. We find that model membrane systems in microemulsion and Ising critical states can mediate roughly equal levels of kinase recruitment (binding energy ∼ -0.6 kB T), whereas a membrane near a tricritical point can mediate a much stronger kinase recruitment (-1.7 kB T). By comparing several models for lipid heterogeneity within a single theoretical framework, this work points to testable differences between existing models. We also suggest the tricritical point as a new possibility for the basis of membrane domains that facilitate preferential partitioning of signaling components.