Random-effects probit model

I am currently using a mixed binomial model with the following specification in a paper I recently submitted (using lme4):

m1<-glmer(y~X1*X2*X3+(1|Subject.ID),data=data,family="binomial")


X1 and X2 are binary predictors

However, a reviewer has asked me to also include a random-effects probit model for the same analysis. What is the difference between a mixed effects model and a random effects model? Am I correct in thinking that a random-effects model would have no fixed effects and would thus be specified by:

m1<-glmer(y~1+(1|Subject.ID)+(1|X1)+(1|X2)+(1|X1:X2),data=data,family=binomial(link=probit))


If so how do I interpret the output. All I get is the Std.Dev. for each random effect.

Alternatively does it refer to a model where I would have a random slope for each of my predictors like:

m1<-glmer(y~1+(X1|Subject.ID)+(X2|Subject.ID)+(X1:X2|Subject.ID),data=data,family=binomial(link=probit))


m1<-glmer(y~X1*X2*X3+(1|Subject.ID),data=data,family=binomial(link="probit"))

as in your case only intercept is allowed to be random. I would make sure though, by searching whether random effects probit model can really be estimated with glmer, there might be some caveats related to exogeneity of regressors, which are ignored in glmer case.