Not always.The source for the data on the sites mentioned in this thread are the various state repeater coordination organizations...
The ARRL has worked out a deal with RFinder in which coordinators can submit their data to RFinder who will then put that data on their web site and RFinder will compile the data for the next edition of the ARRL Repeater Directory. But, many coordinating bodies are not participating in this because of past bad experiences with RFinder. RepeaterBook has state "coordinators" who edit the data for a particular state. Whether or not those "coordinators" get their data from their state coordinating body is probably a state-by-state thing.
Most amateur radio repeater coordinating bodies copyright their repeater listings and simply lifting that data for re-publication elsewhere could be a copyright violation. Some coordinating bodies, such as the South East Repeater Association do not publish their data on the Internet at all. Some coordinating bodies "salt" their published data with fake listings so they can prove copyright infringement much in the same way that Rand McNally used to put fake town names on their printed maps.
Going directly to the source of the data is a good idea for the reasons stated. That doesn't mean that the coordinating body always has the most accurate data. It all depends on how diligent the coordinators are about keeping their databases "clean". I can speak from experience that getting accurate and timely information from repeater owners and trustees is a challenge....so I generally go directly to those sites to find their database page. It's going to be the most current info in my experience.