I usually make the following difference between a process and a lifecycle, based on the state/transition diagram and Object Oriented technoogy.
A process is a suite of interrelated activities that create or transform a "product". A product may be a request document, a written document, a mechanichal part, a software, and so on.
The life cycle describes the successive states of the product as it is transformed by one or several successive processes, from an initial to a finished state.
In the software industry, the "software development process" transform a set of requirements into an executable application. Just identify the software as a product and transform it from its initial state "requirements" into a finished state "application". You have intermediate state like "designed", "coded", "tested",...
If you represent that in a diagram, with boxes and arrows linking the boxes, the process diagram represents activities in the boxes and the product in specific states in arrows. The lifecycle diagram repersents product states in boxes and transformation activities in arrows.
In the software methodology litterature unfortunately, when you see a "waterfall lifecycle", it's generally the description of a process, not a life cycle. (at least in my opinion).
To answer your question on the term "model", it is the description of a generic "process" or "lifecycle" that is customized to represent your particular needs. You plan your activities with one model or the other but you keep the model in mind.
Does it help?