Are there Petri net models with "checking" rather than "taking" transitions?
What you are looking for with "checking edges" are known as Petri nets with read arcs.  In your question you also mention nets with a single read edge, those are called communication-free.  It would be difficult (for me) to point to the original author(s) for these notions, but they are fairly standard.  Relationships between communication-free Petri nets and Parikh images of context-free languages are investigated for instance by Javier Esparza, Petri Nets, Commutative Context-Free Grammars, and Basic Parallel Processes, FCT 1995, LNCS 965, pp. 221--232, and Fundamenta Informatica 31(1):13--25, 1997.