A google search shows much more results for Wohnsitz then for Domizil. I tried to find a good explanation, but most are just like: Domizil is a Wohnsitz.
Regarding to your question in legal context, I will add some further information about Wohnsitz
Ort, an dem eine Person den Mittelpunkt ihrer Lebensbeziehungen hat (§§ 7 ff. BGB). Kinder teilen i.d.R. den Wohnsitz der Eltern (bei verschiedenen Wohnsitzen des Elternteils, der das Kind in persönlichen Angelegenheiten vertritt), auch wenn sie sich tatsächlich an einem anderen Ort aufhalten.
Nach dem Wohnsitz bestimmt sich v.a. der Gerichtsstand und der Erfüllungsort.
Translation (in my broken english): Wohnsitz is the location where a person spend the most time of his life (§§ 7 ff. BGB). Children's Wohnsitz is usually the Wohnsitz of their parents [...][*]. The court of jurisdiction is appointed to their Wohnsitz.
So, I never read the §§ 7 ff. BGB, but there you find all regulations belonging to Wohnsitz. Wohnsitz is a very formal word, used in many official documents, in court, etc. You can also hear the word in news (have a look on elena's example, this could be a typically sentence in a newspaper, but also written in a protocol in court)
Here I want to c&p the sentence from Hendrik Vogt (mich mit fremden Federn schmücken):
You'll hear it in TV magazines reporting about some celeb's new villa.
The word is bumptious. Don't use it.
- Residenz is used, when you name the domicile of an important person or the official residence of an ambassador. Another possibility in this context is Sitz.
- Heim is a used for nursing home, children's home,... You can find it in informal use, like: Ich gehe heim, Das mache ich daheim, ... But note, in this use it isn't a noun.
- Bude is very colloquial, used e.g. by student's, talking about their Wohnheim.
There a handful more synonyms, but I do not want to mention all of them.
Finally, take a look at ngram for Domizil vs Wohnsitz.
[*] I omit the bracket part. It's about children's Wohnsitz if parents home is separate.
Regarding your update, I will add some further thoughts:
Your Wohnsitz is registered from the time when you are born. The Wohnsitz is the same place as your parents home. This is also called as your Hauptwohnsitz (principal residence). When you move to another town, you change your (Haupt)Wohnsitz. If you have two homes, e.g. your parents' home and your student Wohnheim, then you register a Nebenwohnsitz (secondary residence).
Domizil is just another word for Wohnsitz. I think you can use is for your principal and secondary residence. At least in the sense as John Smithers mentioned.