While it is possible to find Unexplored systems within 1000 LY of Sol - a year ago I found an Unexplored Earth-like just 800 LY from Sol - it's by no means common or normal. You've got to go out to 2000 LY before it becomes "common". Out beyond 5000 LY, it's finding another Explored system that becomes scarce - unless, of course, you're traveling on a heavily trafficked route.
As for discrepancies between EDSM and the in-game discovery records, it's because EDSM does not have access to any actual in-game list of discovered planets. There is no such in-game database. Plus, it works both ways. You can get systems that are "discovered" in EDSM, but are Untagged in-game. This might happen if an explorer using EDSM is still out on an ultra-long-haul exploration mission and hasn't come back to port to sell their data yet - in which case, you'd be unwittingly "stealing" their Tags. Or, it could be the case that the EDSM-user blew up and lost all their data, and has never gone back to re-Tag it. This is because EDSM records discoveries in real time, as they are made (unless you've switched that function off in EDSM), whereas the game only records "First Discovered" when the data is sold.
There's no way, either in game or via third parties, to be absolutely sure you're the "first person" to ever see an Unexplored system - all you know for sure is that no-one else has ever sold the data for it yet.