Hola John

Glad to hear you are enjoying Kwiziq!

You are right about the meaning of these two verbs but they are actually interchangeable sometimes. Meter definitely means to "put inside [something]", e.g "Metí los calcetines en el cajón" (I put the socks inside the drawer). Poner means to "place" [anywhere], so you could either say "Puse el vaso sobre la mesa" (I put the glass on the table) or "Puse los calcetines en/dentro del cajón" (I put the socks in/inside the drawer). The preposition with poner will dictate "where" you put the thing. We can say "poner" is more versatile than "meter". Verb meter is specific to mean to put inside, as in "introduce". 

In reference to the sentence from the dictation you mentioned: "las puse en agua", referring to the flowers, we could have used "las metí en agua", meaning exactly the same as you are talking about "introducing" them inside a jug/vase full of water; however the expression "poner en agua" is more commonly used in Spanish to talk about flowers. 

Hope this helps,

Gracias por tu pregunta y un saludo

Inma