You have to understand that, under the gospel of the kingdom, or the gospel of the circumcision, salvation and rewards go together. So yes, your first paragraph is correct.
What Jesus shared in the parables of the kingdom, such as those in the latter half of Matthew, all indicated that. The clearest of course is the sheep and the goats in Matthew 25:31-46, but there are also parable of the talents too.
So when John wrote 1 John, he is writing to the circumcised, as agreed in Galatians 2:9.
But under the gospel of the uncircumcision, or the gospel of the grace of God which Paul preach to us, salvation and rewards are delinked. Salvation is a free gift, but rewards must be earned. That is what Paul was trying to explain in 1 Cor 3
10 According to the grace of God which is given unto me, as a wise masterbuilder, I have laid the foundation, and another buildeth thereon. But let every man take heed how he buildeth thereupon.
11 For other foundation can no man lay than that is laid, which is Jesus Christ.
12 Now if any man build upon this foundation gold, silver, precious stones, wood, hay, stubble;
13 Every man's work shall be made manifest: for the day shall declare it, because it shall be revealed by fire; and the fire shall try every man's work of what sort it is.
14 If any man's work abide which he hath built thereupon, he shall receive a reward.
15 If any man's work shall be burned, he shall suffer loss: but he himself shall be saved; yet so as by fire.
So long as one is able to rightly divide the word of truth, he will understand that rewards are based on building on the proper foundation, which is the foundation Paul laid for us in the Body of Christ.
We in the Body of Christ will also be able to read passages in 1 John 2:3-4, and understand them literally.