What exactly is the 'online' in OLAP and OLTP?
It is simply a remnant of olden times, when it was used in contrast to batch processing. "Online" here means "interactive", that is, requests to the database are processed as they come and responses are given more or less immediately, or at least as soon as they are available. Batch processing would collect requests into, well, batches, and execute them on schedule; responses would be given after the entire batch execution (e.g. next morning). 

Abbreviations OLAP and OLTP hint at another historical artifact:  "on-line" used to be the more common spelling until mid-1980s.