The J-1 visa is a scholar visa for the exchange of ideas. "The exchange of research scholars promotes the exchange of ideas, research, mutual enrichment and linkages between research and academic institutions in the United States and foreign countries."
The H-1b visa is a technical worker visa allowing someone to work in the US because they have skills or technical knowledge that, at least in theory, are not readily available among US workers.
I agree that there can be overlap between the two. But it the university brought you in under an H-1b visa, they have classified you as a technical worker and not a scholar. The only way to dispute that is to ask them to reclassify you. Otherwise, I suggest you seek qualified legal advice.
