Instead of turning this into an academic debate about copyright, a more practical thing to do is to examine the terms and conditions of whatever API you are using. Because if you are going to end up in a conflict with a search API provider, those probably spell out pretty clearly what the provider wants to allow or not and what you are agreeing to by using their API.
Caching is a problem with many geocoding APIs (which I happen to be familiar with) and a good reason to prefer e.g. Opencage over the Google or Here geocoders because unlike most geocoder terms and conditions, Opencage actually encourages you to cache and store things; because it's all open data. The Here geocoder requires you to tell them how much data you store and will try to charge you extra for the privilege of storing and keeping data around. Because it's their data and the conditions under which they license it to you are limiting what you can and cannot do. Search APIs are very similar. Technically geocoding is a form of search (given a query, return a list of stuff).
Caching is a problem with many geocoding APIs (which I happen to be familiar with) and a good reason to prefer e.g. Opencage over the Google or Here geocoders because unlike most geocoder terms and conditions, Opencage actually encourages you to cache and store things; because it's all open data. The Here geocoder requires you to tell them how much data you store and will try to charge you extra for the privilege of storing and keeping data around. Because it's their data and the conditions under which they license it to you are limiting what you can and cannot do. Search APIs are very similar. Technically geocoding is a form of search (given a query, return a list of stuff).