I think the definitions of inhibitory and excitatory neurons are currently defined as a matter of convenience based on observations made by synaptic staining, electrical feature and its morphology.
In other words, neurons with more inhibitory synapses are inhibitory neurons, and neurons with more excitatory synapses are excitatory neurons, and so on.
By detecting inhibitory and excitatory synaptic-specific staining in different colors on the same section, we may be able to determine whether all synapses in a single neuron are inhibitory or excitatory.
Also, I think there are events happening in the process of neural differentiation that distinguish excitability/inhibition, but we don't seem to know how they are happening.
It may be determined by signals from neighboring neurons, Ca++ concentration, etc.
If, at the neuronal precursor stage, the differentiation mechanism that determines excitability/inhibition could be confirmed at the level of gene expression, it might explain why inhibitory neurons only make inhibitory synapses.