The British Officer who was a representative of the Governor General and who lived in a State which was not under direct British rule was called:

The British official who served as the representative of the Governor-General in a Princely State not under direct British rule was known as the Resident. While British India was divided into provinces under direct control, the Princely States were ruled by local princes who accepted British paramountcy or suzerainty [4]. Under the system of subsidiary alliances, these rulers were required to accept a British Resident at their court [2]. The Resident was a senior official and diplomat responsible for managing relations between the British Government and the state, ensuring the ruler adhered to their alliance and maintaining indirect rule [2]. In some larger regions or groups of states, this representative was also referred to as the Agent to the Governor-General (AGG) [3]. However, the specific officer living within a state's capital to oversee its internal and external alignment was primarily designated as the Resident [2].

Sources

1. [3] https://delhihighcourt.nic.in/files/Ebook/relations_0.pdf
2. [4] https://www.jstor.org/stable/2141033
3. [2] https://en.wikipedia.org/wiki/Residencies_of_British_India