Frankly, you won't find a completely correct website out there...those crowd-sourced websites are a good place for you to start though...they usually have some concepts listed up at the top that explain tactics for use in "countering" a champion...you'll have to look at those concepts and decide...are these concepts useful during lane phase? For example, you don't counter Nid during lane phase...she's hard to gank, she can farm even when she can't duel and she has incredible sustain...the traditional counter to poke teams is hard engage...so some sites will list Amumu as a counter...but Amumu can't lane against her (some sites won't list him at all because people only consider laning counters...something to think about there too). He counters her once teamfights start because he allows your team to engage quickly on the other team without taking poke. This kind of stuff is just stuff you have to think through. Counter sites are good to start from, but you have to think through the reasons why someone voted that champion in...maybe that's not the champion you need to pick for laning phase...
That's the kind of stuff he means..."countering" is a much more complex concept than just picking a single champion to counter another champion.