The word "mindfulness" is a rough English translation of the Pali word sati. The Satipaṭṭhāna Sutta is the most influential text on what in the west is known as mindfulness mediation, so perhaps you'd like to start there. I can recommend the commentary Satipaṭṭhāna: The Direct Path to Realization by Bhikkhu Anālayo. Because Anālayo has waived royalty payments from this book as an act of Dhammadāna, you should be able to find a copy easily.
To give you my own answer to your question: First, you say, "we always focus on something". In truth we rarely achieve concentrated focus that tunes out everything else. We rarely "just sit peacefully and observe .. thoughts", though we might think we do, we are constantly evaluating and making judgements about our thoughts. So meditation practice can help refine the skills of focusing and observing. Second, everything you say about not needing a separate concept or word is true in an intellectual sense, but to actually know and experience mindfulness is different from merely having a rational understanding of the ideas.
To give you my own answer to your question: First, you say, "we always focus on something". In truth we rarely achieve concentrated focus that tunes out everything else. We rarely "just sit peacefully and observe .. thoughts", though we might think we do, we are constantly evaluating and making judgements about our thoughts. So meditation practice can help refine the skills of focusing and observing. Second, everything you say about not needing a separate concept or word is true in an intellectual sense, but to actually know and experience mindfulness is different from merely having a rational understanding of the ideas.