As is known, Islam is it's own politics, and we can split politics into two categories, politics which is in accordance with Sharia (سياسة شرعية) and politics not in accordance with the Sharia (سياسة غير شرعية). I am well aware of the different definitions of politics given by the non-Muslims. My question is what is the definition of politics in Islam? And what have the scholars (specifically those that specialize in this area of Fiqh) said in regards to this?
Basically the definition of politics in Islam is not so much of out of Islam. But the source of laws for political laws are different. In Islam the sources of political laws are the Qur'an and authentic hadith. But in out of Islam the source of political laws are different theories made by humans. In fact in Islam the source of Islamic laws are God and out of Islam the source of political laws are humans (in fact Satan).
This is the definition of politics in Islamic view by Ayatollah Muhammad Taqi Misbah Yazdi:
Politics means the method of administering or organizing the society in such a manner that its interests and desires are realized. In simpler terms, politics means the rule of statecraft. What we mean by politics is not ‘real politic’, the concept with negative connotations, which is linked with chicanery, trickery, scam, and deception.
The social laws include civil, penal, commercial, political and international laws. Once we take a glance at the Qur’an, we discover that all kinds of laws in the world for the administration of society, and the management of international relations can be found therein. Apart from the fact that civil laws; decrees on marriage and divorce; laws on trade, transactions, mortgage, loan and the like can be found in the Qur’an (which proves that issues on statecraft, like enactment and presentation of laws for the administration of society are taken into account in Islam), a special right for the Prophet (s) has been stipulated in the Qur’an to enact laws and decrees on particular cases based on circumstances of time and space, and the faithful are duty-bound to act upon the laws issued by the Messenger of Allah (s):
وَمَا كَانَ لِمُؤْمِنٍ وَلاَ مُؤْمِنَةٍ إِذَا قَضَى اللَّهُ وَرَسُولُهُ أَمْرًا أَن يَكُونَ لَهُمُ الْخِيَرَةُ مِنْ أَمْرِهِمْ
“A faithful man or woman may not, when Allah and His Apostle have decided on a matter, have any option in their matter.” (33:36)
In this noble verse, the faithful have been deprived of the option to violate the decision of God and His Prophet (s).
Thus, apart from the orders of God and fixed divine laws, laws enacted by the Prophet (s) are also binding on all those living within the jurisdiction of the Islamic government.
And this is another definition of politics based on views of Imam Khomeini:
Politics means guiding the society and showing the way; it means considering all the interests of the society, and taking into account all aspects of man and society and guiding them toward the things that are best for them, for the nation and the individuals. This is exclusive to the prophets (‘a).
The Qur'an says when humans does not follow laws of God indeed they will be trapped by Satan and their political laws will be determined by Taqoot:
Allah is the ally of those who believe. He brings them out from darknesses into the light. And those who disbelieve - their allies are Taghut. They take them out of the light into darkness. Those are the companions of the Fire; they will abide eternally therein. (2:257)
Islamic Political Theory (Legislation): Volume 1 > Definition of politics and the three powers in Islam by Ayatullah Muhammad Taqi Misbah Yazdi
Imam Khomeini, Ethics and Politics > The meaning of politics and the status of power by Sayyid Hasan Islami
I think your approach to politics and Islam is not right. Islam does not define a political system and it is not concerned with any particular type of system.
Islam came to purify human beings. This purification is required in personal as well as social matters. This purification is required in thought, speech, and actions. Some things are common among humans, like bodily cleanliness. Also, humans play different roles, like father, mother, employee, son etc. Islam addresses how one can purify yourself, within the context of these roles. These instructions can be both positive (like do this) and negative (don't do this).
A human being can also act as a role of "politician". In any definition of it, Islam will address how a politician should think and act and what he shouldn't do or say.
Islam will NOT encompass all the aspects of politics (or even try to define it), but only those, that relate to purification of this role. e.g. one guidance related to politics is "mutual consultation"
And those who have responded to their lord and established prayer and whose affair is [determined by] consultation among themselves, and from what We have provided them, they spend.