Most searches in other websites are just word searches. However EYB also allows you to include filters - for ingredients and categories such as recipe type, ethnicity, specil diet, occasion and course. For that reason the delete search x is underneath the search bar so as to clear the word search and the filters.
Although you want to preserve your search between a Bookshelf search and a Library search, other members may well find that feature irritating as they have to clear the search terms when they wanted to do a completely different search. I would be interested to hear other members views.