How is "firing together" detected in Hebbian learning?
The dynamics of Hebbian or associative synaptic plasticity are thought to be governed primarily through the function of the NMDA glutamate receptor. The short answer is that you are basically right on all counts.


Figure of the NMDA receptor present the nervous system.
By Blanca Piedrafita [CC BY-SA 1.0 or CC BY-SA 2.5], via Wikimedia Commons

The NMDA receptor is a non-selective cation ionotropic receptor. The reversal potential of the NMDA receptor is around 0 mV, in contrast to the resting membrane potential of a 'typical' excitatory neuron which is around -70 mV. This means that a pretty strong stimulus is required in order for the NMDA receptor to be activated. Whereas some current can pass through the channel upon binding of its preferred ligands (glutamate or glycine), the pore is normally blocked by a magnesium ion. This mechanism confers some degree of voltage-dependent regulation of ion flow through the channel. When a strong stimulus raises the membrane potential at sufficient levels the magnesium block is lifted, allowing more current to pass through. The slow kinetics of the NMDA receptor allow the temporal integration of incoming signals. This makes the NMDA receptor function as a coincidence detector or AND gate. There are many more known details about the mechanisms involved but the above should suffice to answer your specific question.

Further reading: https://en.wikipedia.org/wiki/NMDA_receptor