"Puts down the book" strikes me as somewhat unnatural word order. I am not confident that it is grammatically incorrect, but I wouldn't expect a native speaker to use that construction.
You should be able to translate
legger as either "lay(s)" or "put(s)," because these words all require a direct object (e.g., "boken").
However, "lie(s)" is not an acceptable translation because "lie(s)" is intransitive and cannot take a direct object.
In other words, you actively lay (
legger) something on a surface:
Jeg legger et brev på bordet= I lay a letter on the table.
Objects and people passively rest and lie (
ligger) on a surface:
Brevet ligger på bordet= The letter lies on the table.