Rather than the difference arising from Matthew 6:9 alone, I think the difference in understanding is the result of people being familiar with Matthew's version over Luke's or vice versa.
Matt 6:9 (KJV) After this manner therefore pray ye: Our Father which art in heaven...
Luke 11:2 (KJV) And he said unto them, When ye pray, say, Our Father which art in heaven...
Going just by Matthew you would take it to be a template or model.
Going just by Luke you would take it to be a form to be prayed in those exact words.
Combining the two together, you would take it that both are allowable and advisable.
There's no reason to suppose that Jesus meant one over the other rather than both. After all, Jesus himself is found praying in the gospels using different words than those.
Yet he apparently taught this same prayer on two different occasions: In Luke it is in response to the disciples asking him to teach them how to pray. In Matthew it is part of the Sermon on the Mount. It is quite likely, therefore, that on one occasion he said "Pray like this" and on the other "When you pray, say." And if the giving of the prayer to the disciples in response to their request to learn to pray were the earlier occasion, it would make sense that he might have them pray those exact words (since they were novices in prayer). But later, when they were more familiar with prayer, he might teach "pray like this" meaning that they should now begin to compose their own prayers on this model.
And the disciples are found praying in different words in Acts. So obviously they didn't interpret it as meaning that one must always use those exact words. But it would be absurd to suggest that one should never use those exact words. Just because something is a model or template doesn't mean you always have to deviate from the template and can never use the template as is.
As to the prayer being a model, it can be interpreted that Jesus is trying to get them to use chiastic structure in their prayers (at least in those manuscripts of Matthew that include the doxology). Since it follow the structure:
(1) Praise of God (Our father in heaven, hallowed be thy name)
(2) Spiritual requests (thy kingdom come, thy will be done...)
(3) Physical requests (give us this day our daily bread)
(2) Spiritual requests (forgive us our sins...lead us not into temptation...)
(1) Praise of God (For thine is the kingdom, the power, and the glory, forever. Amen.)