Onsite Link Verdict checks the links on a WordPress site and only calls one broken when it can prove it.
A link is reported broken in two cases: the server says the page is gone, 404 or 410, and repeats it on a second request by a different method; or nothing answers, three checks in a row. Everything else goes to a separate tab called Couldn't verify, with the reason in plain English. Servers answer a script differently from the way they answer a browser, and that difference is named for what it is instead of counted as a dead link.
It runs on your own server. No account, no cloud, no credits. Nothing runs while a visitor loads a page, and every pass has hard ceilings in the code: 100 links, 45 seconds, 2 KB read per response, one request per unique URL.
The free version finds and reports. Pro replaces a URL across every post in one pass, after showing the full list, with an undo, plus CSV export and a silence list.
Free on the WordPress plugin directory. Pro from 29 EUR a year.