Spike Timing Dependent Plasticity interpretation
The best explanation for STDP that I've ever seen is Nicky Case's Neurotic Neurons. In this Explorable Explanation, I think you'll find that your misunderstanding lies in the fact that both the pre-synaptic (Neuron A) and the post-synaptic neuron (Neuron B) exist in a network of other neurons. They are not only connected to each other. You are correct in saying that it often takes more than one incoming spike to make a neuron fire. These spikes are usually coming from other neurons (Neuron C, D, E... etc). So it's incorrect to think that "the only way a postsynaptic spike occurs is when the presynaptic neuron fires into the postsynaptic one".

So, to summarize, yes the time interval is the time difference between the last pre-synaptic spike to the first spike of the post-synaptic spike. These neurons exist in a network of neurons. It's probably best to think of STDP as a coincidence detector.