"In the case where" would be used to refer to a particular or specific situation. This is because the is a definite article. You are correct in believing that "In a case where" would be used to refer to a general situation. This is because a is an indefinite article. There is no distinction here between formal and informal settings. More @Wikipedia
Hover over any link to get a description of the article. Please note that search keywords are sometimes hidden within the full article and don't appear in the description or title.