It could mean:
1. Some (and only some) will know Him, and others will be able to believe on their words (but will not know Him).
2. Some, initially less than all, will know Him, and others will, initially, believe on their words. But if the others who believe on their words follow the same path as those who know Him, they will also grow to know Him as well.
The correct choice between these two is described in Nephi's account where he could not believe his father, Lehi. Then he prayed and the Lord "visited" him by softening his heart so he could believe his father's words. Then he developed faith to receive stronger impressions, and acted consistent with them. Then he was able to "hear" the Lord by continuing on that path. Finally he had angels minister to him and prepare him to receive an audience with the Lord. And, after remaining true and faithful to the path, he at last received an audience with the Lord.
Nephi's spiritual development is described in detail in the early chapters of The Second Comforter: Conversing with the Lord Through the Veil.
D&C 93:1 says "every soul" not just a few. Not just a select group. But "every soul." I believe it means all. Not just a few; while others are relegated to believing on their words.
D&C 46: 14 is talking about where people begin. Not where they finish.