"Cheers! Drink more!"
The connotations are subtly different. 多喝一点 is more cheerful encouragement, where is 喝多一点 is more serious.
This way of translating is what confuses beginners. It asks to translate 'Drink more', not 'Drink a little more'. Beginners do not know the customs and colloquialisms, so we do not think of changing the meaning slightly.