There’s three major ways that come to mind how different neurotransmitters convey different information. The first is one being excitatory, and the other being inhibitory. That’s pretty straightforward, and there are examples all over the nervous system.
The second would be different receptor kinetics. For example, if one receptor activates and deactivates (or inactivates) quickly, and the other slowly, then the first might convey more temporally precise or phasic information, while the other might convey more tonic information about presynaptic activity. To be honest, I have trouble thinking of examples of this for ionotropic receptors. AMPA-type glutamate receptors are fast, and nicotinic ACh receptors are somewhat slower, but I’m not sure that is so dramatically different. Also, I’m not sure how often those receptors are found on the same neuron. There are probably examples in the retina?
The third way would be through different receptor actions. Your example is just right for this, because glutamate can act on ionotropic receptors, but last I heard, dopamine receptors are exclusively metabotropic, so their electrical effect is through activating or inactivating other channels, which affects excitability. Metabotropic receptor pathways tend to act more slowly, and they can have diverse effects. For example, even if they don’t drive spiking themselves, they can shift resting potential or input resistance to influence the likelihood of spiking following ionotropic receptor activation, and those effects take a while to subside. I don’t have the details at my fingertips, but that is probably how excitatory D1 receptors on medium spiny neurons in the direct pathway of the striatum work. There are examples of this throughout the nervous system.
One could probably write whole books on this subject!
Neurotransmitters are unique in their actions - excitatory (like acetyl-choline, epinephrine etc.); inhibitory (endorphin, GABA, serotonin); even both (dopamine); and also may be modulatory. Besides their role in neurotransmission, they play important roles in depression, migraine, anxiety. anaphylaxis, cardiac function, asthma, memory, learning, behavior, even Alzheimer's disease etc.