About definition of model In Model theory, the definition of a model is a set. 
Can it be a proper class?
ZFC has a model and maybe some models is a proper class.
Definition of a model needs to include a proper class.
Is it correct? 
Id like to ask your opinion.
 A: A model is, by definition, a set. So no, "class models" are not models in the usual sense. Lots of theories besides ZFC also have "class models," but that's a different sort of thing - and really can't be nicely studied without working in a class theory like NBG or MK.
A: Normally a model is understood to be a set, but it is not rare for people to consider proper class models. Sometimes they do this informally with no rigorous justification.  One way to formalize it is in Godel-Bernays set theory where proper classes actually exist. 
ZFC has models if it is consistent (and pretty much everyone who cares believes that ZFC is consistent).  And, because ZFC is a first order theory, if it is consistent then it has set models, indeed it has countable models.  People do sometimes consider proper class sized models of ZFC too -- which can be formalized in various ways.  One way is to use Godel Bernays, of course.  A more common way is to use "definable classes," as described in difference between class, set , family and collection and at a higher level at difference between class, set , family and collection 
A: Assuming the existence of a class-size model of a theory is very common. Sometimes model theorists want to talk about how "large" saturated models work over a "small" set of parameters. There a formal notions that define "large" and "small" when talking about set sized models (e.g. Hodges's $\kappa-$bigness). However, sometimes these details clutter the big picture. So, it isn't uncommon for people to assume that there exists a class size model $\mathbb{M}$ (often referred to as a monster model - but this term can also just mean "large enough") where "large" means "a class sized subset of $\mathbb{M}$" and "small" means "a set sized subset of $\mathbb{M}$". 
