If I'm not wrong both "quasi" and "pseudo" denote the same thing, namely the optimization under wrong distributional assumptions. Moreover I think that the terms are not restricted to the assumption of normality. Is there an experienced reader who can confirm this? Cheers!
Quasi-likelihood and Pseudo-likelihood mean different things. If the probability model is possibly misspecified, then the likelihood function is called a quasi-likelihood function (see White 1982 econometrica for example). In the special case where the probability model is correctly specified then the quasi-maximum likelihood estimation is the same as maximum likelihood estimation. The terminology "pseudo-likelihood" is not as established but typically means that independence assumptions are violated so that the the independence assumptions which permit the likelihood function to be constructed as a product of other likelihood functions are violated but the likelihood function is constructed as a product of other likelihood functions anyway. Thus, every pseudo-likelihood function is a quasi-likelihood function but every quasi-likelihood function is not necessarily a pseudo-likelihood function. See Besag 1986 "Analysis of Dirty Pictures" (Journal of Royal Statistical Society Series B, Vol. 48 for discussion of the pseudolikelihood function). These terms are not restricted to the assumption of normality.
The two names denote the same concept. See f.e. Hayashi's Econometrics. Concerning the second part I am not sure, but, normally, it is explained as the estimation if the model is misspecified (generally) - "in ways that do not affect the consistency of the estimator" (Davidson, Mackinnon - Econometric Theory and Methods.