Comparing "Lethal Counteratttack" and "A Lannister Pay His Debts", " I Never Bet Against My Family" doesn't say "One of your", so does this mean that I can kneel a "Lannister" character controlled by my opponent to pay for the effect?

No. Kneeling a Lannister character is part of the cost of the effect, and costs can only be paid with cards you control.

In any "do X to do Y" construction, X is always the cost.