I think that, if a model's outcome variable has only two categories, then the appropriate term for the analysis is binary logistic regression, regardless of the number and type of predictors. For details on performing actual logistic regression analysis, please see links in my recent related CV answer as well as this tutorial, this chapter (SPSS examples) and this nice set of slides (Stata examples).
In regard to potential multicollinearity, if you're not familiar with this topic, I suggest you start by reading corresponding Wikipedia article. Also, check some CV discussions, for example this, this and this. If you have access to paywalled peer-review journals through your institution, you might want to check this paper and this paper. This blog post presents some options on dealing with collinearity. More details and references on the topic of multicollinearity in general (not specific to logistic regression models) can be found in my recent relevant answer. I hope that this is helpful.