Difference between spiking and firing
In your quotes, the terms spiking and firing are used as synonyms. Indeed, both terms refer to the same phenomenon of action potential generation and in this context there is no physiological functional difference between spiking and firing.  

However, note that in some contexts there may be a difference, namely some authors distinguish between a neuron spiking and bursting; the former referring to a tonic firing mode, the latter to a bursting firing mode (e.g. Ramcharan et al. (2000)).

Reference
- Ramcharan et al., Vis Neurosci (2000); 17(1):55-62