The phrase 'Personal Savior' is not found in scripture. And 99% of the time the phrase is used without such an explanation (in my experience), adding a layer of unnecessary verbage that distracts and confuses unbelievers. There are people who treat the phrase as if it should be in the Bible, and as if people aren't saved unless they through the word 'personal' in front of Savior.
My car is my personal property. If you use a slim jim and open the door and drive it around, you can be arrested. But me believing in Jesus does not keep you from believing in Jesus.
There are many words not written in scripture, but that doesn't mean it was never inferred.
John, in chapter 1, verse 1 says, "In the beginning was the Word and the Word was God..."
John never mentions Jesus by name, but it is inferring to Jesus.
This is part of simple discernment of scripture.
In order for one to receive their healing for themselves, the MUST receive the promises concerning healing for themselves, PERSONALLY.
Otherwise, healing will never come.
In like manner, if you don't receive Jesus personally, then you do not have Jesus in your heart, which translates to... you are not born again.
For it is written, "He who has the Son has life, and he who does not have the Son of God does not have life."
Mat 10:40 He that receiveth you receiveth me, and he that receiveth me receiveth him that sent me.
Mat 10:41 He that receiveth a prophet in the name of a prophet shall receive a prophet's reward; and he that receiveth a righteous man in the name of a righteous man shall receive a righteous man's reward.
Joh 12:48 He that rejecteth me, and receiveth not my words, hath one that judgeth him: the word that I have spoken, the same shall judge him in the last day.
Joh 12:49 For I have not spoken of myself; but the Father which sent me, he gave me a commandment, what I should say, and what I should speak.
Joh 12:50 And I know that his commandment is life everlasting: whatsoever I speak therefore, even as the Father said unto me, so I speak.
Joh 14:20 At that day ye shall know that I am in my Father, and ye in me, and I in you.
Joh 14:21 He that hath my commandments, and keepeth them, he it is that loveth me: and he that loveth me shall be loved of my Father, and I will love him, and will manifest myself to him.
Joh 14:22 Judas saith unto him, not Iscariot, Lord, how is it that thou wilt manifest thyself unto us, and not unto the world?
Joh 14:23 Jesus answered and said unto him, If a man love me, he will keep my words: and my Father will love him, and we will come unto him, and make our abode with him.
Joh 14:24 He that loveth me not keepeth not my sayings: and the word which ye hear is not mine, but the Father's which sent me.
It is not possible for Jesus to be in you, if you fail to receive Him for yourself, personally.
No, it is not written that way specifically, but we are no longer children, unable to decipher and understand what is being said.
With a little thought, and very little discernment, it is easy to see, we receiving Jesus personally, by accepting the word of God or His promises, by faith. For we are born again BY and THROUGH the word of God.