First of all, I would for sure not quit your job unless you are sure you will receive the resident permit for self-employed work. Expect it to be a long process, depending on what kind of residence permit you have currently.
Also be prepared to deal with one specific problem:
German law knows the co-called "Freie Berufe" (which is meant by "liberal professions" in the link you provided. For most government officials in Germany this means medical doctors, lawyers and possibly architects, not necessarily "IT people".
When it comes to IT people (I am one myself) you will find the term "Engineer" in the official list of "liberal professions" and will probably argue that what you are doing is "like engineering". And you may be right with that, but you will have to proof it and explain it to people who will possibly not have the least idea what you are talking about and will use terms and ask questions which you may not understand. (And this will not be a German language problem.)
To cut a long story short: A pretty high court has ruled that IT people may be engineers in the sense of the law on liberal professions. But you will at least have to demonstrate that you have gone through an education which is at least similar to that of an engineer, which usually means a university diploma.
And a final word from experience: Unless your experience is very specific (like certified special aspects of SAP or similar) 700 EUR / day sound a lot to me. Expect rather 500 and you will not be disappointed in the end. Please also understand that this will be a gross income only for the days you will be working for a customer and you will have to cover all of your taxes and social security, pension plan and whatever other cost yourself. If you sell 180+ days / year at 500 EUR / day you will possibly be better of than the salary you receive now, but it's not as profitable as it may sound in the first place.