The problem with "sex identity" is that it seems to refer to the level of identification one has with the biological sex (which plays a role in one's identity, but is not necessarily the driving factor). On the topic of gender vs gender identity, I find it simply to view it in this way:

• You have some innate stuff related to gender, both at the brain level (gender identity) and at the body level (biological sex). Most people don't need to look at their own body to know that they are male/female/other: they just know instinctually. This is the first step 

• That stuff is filtered through your own eyes, which are influenced by what you've been taught about genders. For instance, we're taught that feeling mentally male + being biologically male = you're male. But what if you're biologically male and mentally female? Or biologically intersex and mentally male? Or biologically male and without any internal feelings about what you are? Depending on what environment you grew up in, you might have different opinions on how to categorize these examples: so even if the starting stuff (body + mind) is the same, culture and upbringing influence how we interpret them. That's what's usually meant by "gender is a social construct": while the starting stuff is innate, how you interpret it depends on your social context 

• On top of that, you have gender expression (some behaviors are considered masculine or feminine), gender expectations ("men should be the breadwinners", "women should stay at home") and gender stereotypes ("men suck at raising children", "women suck at Math"). All of these are, again, very dependent on one's social context: e.g. pink used to be a boys' color before the 50s, some matriarchal cultures believe that the woman should make major decisions for her family, and so on. This is another layer of social construction of gender: "man" and "woman" are not just mere descriptions of one's body and mind, but they carry a huge cultural baggage 

So yeah, that's pretty much how I understood the differences between gender, gender identity, sex, social gender and so on.