Say that someone asked how secure my password is, if I give them the https://howsecureismypassword.net/ results of it, (especially if I disable the named numbers mode,) can that person brute force with the website's algorithm and possibly find my password?
By posting those results, you are limiting the pool of guesses the person has to use. However, this does not make brute forcing faster in a practical way, it simply limits the guesses the person will try. With a strong password, this brings diminishing returns -- excluding guesses without numbers or symbols doesn't help all too much.
However there is one exception: if the site reports you're using a "commonly used password" then it makes brute forcing extremely fast. And if you're using a password from that list, you deserve it!
There should be no risk. The password you enter on https://howsecureismypassword.net/ is in no way related to where you actually use it. (Or don't tell the one who knows where you use it that you used https://howsecureismypassword.net/ for this :)
Say you test your password that you use on amazon.com ... You can enter that fine on https://howsecureismypassword.net/. Als long as there is no relation to that fact that you're using it on amazon.