What is the definition of a saint? All the early christians were saints? Shouldn't all believers be saints?
Yes - all believers who have surrendered their life to the Lordship of Jesus and have become Born-Again (indwelled by the Holy Spirit of God) are called "saints" throughout the New Testament. So, yes, you are right, this would apply to present believers.
It is the indwelling Holy Spirit of God that thereafter helps us in our daily walk with God, through Christ. Much could be said but here is just one example:
Likewise the Spirit also helpeth our infirmities: for we know not what we should pray for as we ought: but the Spirit itself maketh intercession for us with groanings which cannot be uttered.
And he that searcheth the hearts knoweth what is the mind of the Spirit, because he maketh intercession for the saints according to the will of God.
And we know that all things work together for good to them that love God, to them who are the called according to his purpose. (Romans 8:26-28)
One is not considered a "saint" unless one is born-again, as Jesus said one must be, however, the Biblical definition of "saint" is definitely not the same definition as that of Roman Catholicism.