> we imagine injecting 1 amp of current into a single node
I have always had a problem with statements like this. Current is a (measurable) property of an edge, not a node (vertex). One can’t “inject” current into “a single node."
By definition, an edge connects two nodes. So clearly, "inject current into a node" is just shorthand for "inject current along a (real or hypothetical) edge leading to a node". This is very common terminology in electronics.
Sure, but wouldn’t it be clearer if they talked about a specific edge? Because there can be several edges leading to a node. (Or, are they talking about, say, all of the “inbound” edges taken together?)
Thats equivalent according to KCL - I think the equivalent model is "connect a current source I from ground to a node." Thats equivalent to saying "imagine the sum of all currents through edges was -I"
I imagine you are using an imaginary wire that is perpendicular to the grid and is not shown in the graphics. (Like a multimeter probe.)