to write: to set words or signs down on a surface so they last
Katav covers the whole act of putting language into lasting visible form, from a scribe drawing letters on a scroll to a command cut into stone. The same verb fits whether the writing is private or public, sacred or everyday, on parchment, a tablet, or a doorpost.
The shades below are not different meanings, just the different settings the one act shows up in. The thread running through all of them is permanence: writing is what makes a word stay, so it can be read again, kept as a record, produced as proof, or remembered long after the moment has passed. That is why the Bible so often pairs writing with covenant, law, and witness.
The most loaded uses are the figures of speech built on that permanence: sin written down so it cannot be denied, a name written in a divine register of the living, or instruction written on the heart so it becomes second nature. These all lean on the ordinary sense, that what is written endures and can be brought up again.