My understanding is that a connection between two neurons has a weight, but a neuron itself does not have a weight.
If connection c connects neurons A to B, then c has a weight w, but A and B don't have a weight. w determines if A has a strong influence on B or a weak influence on B.
But this Wikipedia article says: "The connections between artificial neurons are called ‘edges’. Artificial neurons and edges typically have a weight that adjusts as learning proceeds."
The 2nd sentence seems to imply that a neuron has a weight, too.
Does a neuron really have a weight?