Given a weighted graph $G = (V,E)$, the indegree of a node is defined as $ k_{in}(i) = \sum_{j:j \rightarrow i} A_{ji} $ where $A_{ji}$ is the weight of edge from node $j$ to $i$. My question is if there are $m$ nodes pointing to $i$, what's the terminology for $m$? Is it the number of inlinks?
The quantity you are asking about is usually called the "fanin" of node $i$. The dual quantity which counts the number of nodes to which $i$ points is called the "fanout". The terms are standard at least in the circuit design and computer science literature, but I can't seem to find them in graph theory texts at the moment. Searching google for something like directed graph "fan in" will reveal many papers that use the term. 


I've seen papers where $m$ is called the indegree and $k_{in}$ is called something else (such as weighted indegree). Using indegree and fanin as Vel Nias suggests would be fine too. But you need to define it as there isn't universal agreement on how to interpret these names in the weighted graph context. 

