Of Prayers and Vain Repetitions

"Catholics pray in vain repetitions. The Lord said to never do that!" -- Uninformed Non-Catholic

This is likely a reference to:

But when ye pray, use not vain repetitions, as the heathen do: for they think that they shall be heard for their much speaking.
-- Matthew 6:7 (KJV)

Apparently formalized prayer such as the Catholic Church prescribes is the vain repetition that has been prohibited. But did our Lord truly prohibit the practice?

And he left them, and went away again, and prayed the third time, saying the same words.
-- Matthew 26:44 (KJV)

Apparently our Lord has no problem with it when He offers the same prayer more than once. But is that His privilege? Maybe the rest of us are not permitted to offer repeated, formalized prayer?

Did our Lord possibly forbid that?

And he said unto them, When ye pray, say, Our Father which art in heaven, Hallowed be thy name. Thy kingdom come. Thy will be done, as in heaven, so in earth. And forgive us our sins; for we also forgive every one that is indebted to us. And lead us not into temptation; but deliver us from evil.
-- Luke 11:3-4 (KJV)

So our Lord said not to offer vain repetition as prayer. But He prayed repetitiously on at least one occasion and also instructed His apostles to do the same thing with the Our Father prayer. So either our Lord violated His own instructions or else a bunch of Protestants misunderstood His instructions.

Which of those seems more likely to you?

Mind you, the Protestants haven't kept their noses clean about this themselves. I assume they pray the "Our Father" once in a while. Plus, ultimately prayer is praise. So is singing the same song in praise a vain repetition?

"O sing unto the LORD a new song: sing unto the LORD, all the earth."
-- Psalm 96:1 (KJV)

How many times have Protestants sung Amazing Grace?

To the Protestants: There is no need for a double standard; one will do nicely.