The increasing volume of academic literature makes it essential for researchers to organize, compare, and contrast collections of documents. Large language models (LLMs) can support this process by generating schemas defining shared aspects along which to compare papers. However, progress on schema generation has been slow due to: (i) ambiguity in reference-based evaluations, and (ii) lack of editing/refinement methods. Our work is the first to address both issues. First, we present an approach for augmenting unannotated table corpora with \emph{synthesized intents} and apply it to create a dataset for studying schema generation conditioned on a given information need, thus reducing ambiguity. With this dataset, we show how incorporating table intents significantly improves baseline performance in reconstructing reference schemas. Next, we propose several LLM-based schema editing techniques. We start by comprehensively benchmarking several single-shot schema generation methods, including prompted LLM workflows and fine-tuned models, showing that smaller, open-weight models can be fine-tuned to be competitive with state-of-the-art prompted LLMs. Then we demonstrate that our editing techniques can further improve schemas generated by these methods.