This term is specific to the Indian business environment. It doesn't really make sense in isolation, only with respect to its alternative: the promoter-run business. A "promoter" is anyone with a controlling interest in the business, which generally means a significant shareholder. It quite often refers to founders, though a promoter does not need to be a founder. Although founders generally start as promoters, by necessity, they can sell their shares and otherwise transition out of the promoter role.
With this context, we can see that a "professionally managed" corporation is essentially one run by "hired guns", rather than the founders. This means that the C-suite is filled with people who are experts in managing a business, rather than people who are trying to execute a particular personal vision. Of course, the C-suite must execute on the vision provided by the promoters. But in a sense, they do this as a "professional service", rather than because they believe in the vision personally.
The second part of your question is false. Promoters, on average, hold 50% of large corporations in India. If your premise were true, all of these corporations would be "promoter managed" rather than "professionally managed". In fact, many of these corporations are professionally managed. Take, for instance, Caprihans India, which is professionally managed. It had a whopping 75% promoter holding until 2023. Hindustan Unilever is another professionally managed company with 60% promoter holding. There are numerous other examples.