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Definition of begin verb from the Oxford Advanced Learner's Dictionary. [intransitive, transitive] to start doing something; to do the first part of something. Shall I begin? begin at something Let's begin at page 9. begin with something I'd like to begin with a simple question. begin by doing something She began by thanking us all for coming.
When you begin something, you go ahead with its first part or stage — you can begin the day with a good attitude, or begin a very long book over and over again, always falling asleep before you get to chapter two. You also might begin a journey by lacing up your hiking boots.
If there exist..... is the subjunctive form of the verb (about which you will find hundreds of illustrations on the net). It's like saying: If there be another challenger, let him come forward. - a construction that has a somewhat poetic or theatrical ring.
The verb exist is stative and is not usually cast in the progressive form, except as an adjectival participle: it bears a "progressive" sense inherently. You are casting exist in the wrong tense: the teacher is referring not to the present existence of the phrase but to its past existence, before the students used it.
But exist is intransitive, so you cannot form a passive meaning from it, so existed cannot be used in this adjectival way. The "-ing" form, on the other hand, is called a "present participle", and when used adjectivally it is active in meaning, so the existing location makes perfect sense.