Apparently I can't find an answer to this on the internet, maybe because it's an obvious answer but still. Is there a reason to keep your full name disclosed on the internet?
Yes and no. What matters about online security isn't about keeping all personal information about yourself totally tight lipped, but being able to control the amount of personal information you disclose in the right context.
Online security is about being in control of how you present yourself and your online identities, and how to prevent others from being able to link the personas that you want to keep separate.
For example, on sites where you discuss your professional interests and such, you may want to use your real name, to build your personal brand that you can use with employers. On social networking sites where you associate with friends and hope to be searchable by old, lost friends, you may want to use the name you expect those people would be able to search, which is likely going to be your real name. On sites where you discuss your guilty pleasures or where you discuss sensitive political issues from viewpoints that may get you in trouble with friends and families, you may want to use a different identity which can't be tracked back to your real identity and which you should never tell anyone in your real life about. In sites where you discuss things that can get you into trouble with authorities, you may want to decide to use a throwaway identity. On sites where you do online activism, where you do need a persistent identity so people can contact you, you may want to create an anonymous identity.
Depending on how much protection you want and the sensitivity of each personas, partitioning may include using a separate email, and in some circumstances, a separate browser profile or virtual machine, and only used via proxy/tor. You may want to have multiple such identities/personas, depending on how much you want to partition your online identities and how problematic it would be for you if anyone linked those personas together.
To keep your online identities personas from being linked, you would want to learn the techniques that can be used to link up your identities. These are techniques such as Cookies, Supercookies, browser and OS fingerprinting, IP address, drag net surveillance, spyware, etc. Some of them may not be relevant to you, but you would want to learn their capabilities and how to avoid that particular techniques when they're relevant.
Or, you may decide that you want to live your life the same online and offline unashamedly, and just deal with the repercussions with a hard head. That's a valid choice as well, but be sure that when that happen that's your choice and you are in control, rather than a situation where you find yourself helplessly into.