“Why would Paul, who I would think would meet the definition of “saved” need to keep running the race?”
All of these verses explain why Paul was running; not because he feared losing his salvation, but out of love and reward of pleasing his Savior.
2 Cor. 5:14 For the love of Christ constraineth us; because we thus judge, that if one died for all, then were all dead:
1 Cor. 3:12-15 Now if any man build upon this foundation gold, silver, precious stones, wood, hay, stubble; 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. If any man’s work abide which he hath built thereupon, he shall receive a reward. If any man’s work shall be burned, he shall suffer loss: but he himself shall be saved; yet so as by fire.
1Cr 9:24 Know ye not that they which run in a race run all, but one receiveth the prize? So run, that ye may obtain.
Phl 3:14 I press toward the mark for the prize of the high calling of God in Christ Jesus.
1Cor 9:25 And every man that striveth for the mastery is temperate in all things. Now they [do it] to obtain a corruptible crown; but we an incorruptible.
2Ti 4:8 Henceforth there is laid up for me a crown of righteousness, which the Lord, the righteous judge, shall give me at that day: and not to me only, but unto all them also that love his appearing.
Jam 1:12 Blessed [is] the man that endureth temptation: for when he is tried, he shall receive the crown of life, which the Lord hath promised to them that love him.
1Pe 5:4 And when the chief Shepherd shall appear, ye shall receive a crown of glory that fadeth not away.
Rev 2:10 Fear none of those things which thou shalt suffer: behold, the devil shall cast [some] of you into prison, that ye may be tried; and ye shall have tribulation ten days: be thou faithful unto death, and I will give thee a crown of life.
2Ti 4:7-8 I have fought a good fight, I have finished [my] course, I have kept the faith: Henceforth there is laid up for me a crown of righteousness, which the Lord, the righteous judge, shall give me at that day: and not to me only, but unto all them also that love his appearing.
Hbr 12:1 Wherefore seeing we also are compassed about with so great a cloud of witnesses, let us lay aside every weight, and the sin which doth so easily beset [us], and let us run with patience the race that is set before us,
As a child you want to please your parents and not shame the family name. You may be alienated from the family and even disowned but it does not cut off the family genes. You are still biologically the family; so to with the family of God.
Obtain what?