I am just interested in finding another word for sentence finisher in English, or basically someone who has a habit of cutting off another person's sentence and putting in their own statement.
While not a single word, and without any history of use in this context, consider the term conversational usurper, where usurp means
to seize, take over, or appropriate (land, a throne, etc) without authority
I think you mean someone who appropriates your point. If you are making a point and another person breaks in to give his own interpretation of what you are trying to say, he has appropriated your point or your response. Often, the explanation used by an interrupter who takes over the sentence is not in keeping with what the speaker had in mind. That person has appropriated your argument and turned it to his own purpose. You see this in informal political debate sometimes.
Since Merriam Webster defines "interrupter" (or "interruptor") as "one that interrupts" and, the same dictionary, defines "interrupt" (verb) as "to stop or hinder by breaking in 'interrupted the speaker with frequent questions'", you can use "interrupter" (or "interruptor").
Incidentally, you'll find the example sentence that Merriam Webster reports fits the case in the question.
To further illustrate its usage, consider "Interrupters might also just be excited or eager to make their point." (CBS NEWS)
Not a single-word answer but that's because there are none, excluding "interrupter" which kinda gets the message across.
to butt in Verb phrase: butt in, to meddle in the affairs or intrude in the conversation of others; interfere: It was none of his concern, so he didn't butt in.
Macmillan defines butt in as: to join a conversation or activity without being asked to
Neologic solution: a "buttsome" or "buttical" person. (a shame these adjectives do not exist)