Is my local monarch or the British government a bigger influence in my life?
Depends, some smaller states actively copied British India and in other states, the local prince was a bigger influence. In large states the reigning prince was the primary mover and shaker although he would borrow heavily from British India despite personally overseeing the state's affairs.
Hyderabad the largest princely state for example was quite distinct. It followed the Islamic calendar, had its own civil service, currency, the revenue system was a maze of petty chiefs and potentates and all the Nizams left a mark on the city because their rule was deeply personal, the most famous being the last Nizam.
Is the line between princely territory and British colony well defined?
In some case yes, in other areas it was more ambiguous and deliberately so. My source for the following is The Imperial Gazetteer of India vol. IV (1907):
The princely states had no connections to foreign powers, that was the paramount power's bailiwick.
Inhabitants of princely states were subjects of the Prince, foreigners in British India but they also enjoyed the same protections that British subjects did when in foreign lands.
Except in the case of personal jurisdiction over British subjects and residuary jurisdiction, Princes and their subjects were free from the control of the laws of British India. Thus if a criminal wanted in British India, escaped to a princely state, he could not be arrested and handed to the British without the ruler's permission.
The British could intervene if they felt that the internal peace of a princely state was seriously threatened.
The princely states shared in the railways, commerce, ports and markets of the paramount power.
Subjects of a princely state were admitted to most offices of the British government.
There was the doctrine of isolation which meant that the employment of European subjects and Americans in princely states, without the previous sanction of the Government of India, was strictly forbidden.
No prince could receive even the commercial agent of a foreign power at his capital.
The Paramount power was the sole fountain of honour, thus the prince could accept no title, honour, or salute from any foreign court or foreign body and he could not issue passports to his subjects.
Outside their Prince's dominions, all Princely subjects became for all intents and purposes, British subjects.
If a Prince wished to extradite a foreign criminal he would have to petition the Government of India to do so.
All disputes and boundary questions between Princes was referred to the British for settlement.
Princely states maintained a small armed force mainly for internal security and ceremonial purposes. The Indian army defended British and Princely India.
Princely states however had to render effective co operation to the Indian army as when required, and in times of emergency they had to fulfil all the roles assigned to them.
Posts in the interior of princely states could not be fortified nor could princely states construct arms and ammunition factories.
The paramount power controlled the princely states' railways, telegraphs and post.
Princely states had to allow the Indian army to arrest deserters, secure local supplies, occupy cantonments and positions.
The Imperial service troops recruited from Princely subjects were trained, inspected and supplied by the British. They were maintained by some Princely states and were placed at the disposal of the British Government by their rulers. At the beginning of the 20th century they numbered 18,000 men.
The paramount power was liable to intervene in cases of grave misrule.
Certain practices like sati, slavery and barbarous punishments were prohibited across the sub continent. The British did tender advice on matters of general welfare.
Sanads of adoption replaced the doctrine of lapse. The paramount power settled succession disputes.