Step 1: Understanding the Question:
The question tests the fundamental nutritional and metabolic classification of microorganisms based on their energy and carbon source requirements.
Step 2: Detailed Explanation:
• Phototrophic Organisms (Statement I): Phototrophs use light as their source of energy. While "photoautotrophs" use $CO_2$ as their carbon source, the general term phototroph can include organisms that utilize a "principal carbon source" (which could be $CO_2$ for autotrophs or organic compounds for photoheterotrophs) to build their cellular biomass. In many introductory contexts, this is considered a valid general description of their carbon uptake.
• Auxotrophic Organisms (Statement II): An auxotroph is a mutant organism that has lost the ability to synthesize a particular organic compound (such as an amino acid, vitamin, or nitrogenous base) required for its growth. Therefore, it requires these specific growth factors to be added to the minimal medium (the principal carbon and energy source) to survive. This is a fundamental concept in microbial genetics.
Final Answer:
Both statements provide accurate, albeit simplified, definitions of these microbial nutritional groups. Therefore, both are considered correct.