There are a few different possible scenarios that could be spun out of the description in the OP. Let’s look at them separately.
In each scenario, the initial circumstances are the same: man on second, first base open, batter hits the ball into play.
Batter-runner advances to second, base-runner on second doesn’t move.
The base-runner on second, seeing the hit, elects not to advance; he stays put on his base.
The batter-runner rounds first and reaches second.
You now have two runners standing on second base. The fielder tags both. Who is out?
By rule, the original base-runner is entitled to the base, because he reached it safely and was not forced to leave it. The batter-runner is out (and gets yelled at by the manager).
Batter-runner advances to second, base-runner starts advancing but returns.
The base-runner on second, seeing the hit, elects not to advance.
The batter-runner rounds first and heads to second.
The base-runner, seeing him coming, decides to head toward third.
The batter-runner reaches second.
The base-runner changes his mind, turns around, and goes back to second.
Again, you have two runners standing on second base. The fielder tags both. Who is out?
By rule, the batter-runner is now entitled to the base, because he reached it safely. The original base-runner gave up his entitlement to second base when he voluntarily chose to leave it and the batter-runner got there safely behind him. Therefore, the base-runner is out (and the manager yells at him instead).
Batter-runner heads toward second, base-runner starts advancing but returns first.
The base-runner on second, seeing the hit, takes a few steps toward third.
The batter-runner, rounding first, sees the base-runner moving, and heads to second.
The base-runner changes his mind, turns around, and goes back to second before the batter-runner arrives.
Again, you wind up with two runners standing on second base. The fielder tags both. Who is out?
The base-runner on second did not surrender his right to the base simply by stepping off. That happens only if the trailing runner gets to the base first. By returning immediately to the base, he retains his right to it. The batter-runner is out (and the manager yells at both of them).
Batter-runner heads toward second, base-runner advances late to third.
The base-runner on second, seeing the hit, chooses not to advance.
The batter-runner rounds first and heads to second.
The base-runner, seeing this, chooses to head toward third.
The fielding team throws the ball to third base. The fielder catches the ball and steps on the bag.
The base-runner then touches the bag.
Is he out?
This, I believe, is basically the original question from the OP. The answer is no (as stated in other answers above). The base-runner must be tagged for the fielding team to record an out. (And the other team’s manager yells at the third baseman.) The force-out situation does not become applicable until the batter becomes a runner, which happens when the entire play is complete and all results are resolved.
The basis for this distinction might be made clearer by the following.
Batter-runner heads toward second, base-runner feels compelled to advance, then batter-runner changes mind.
Same basic setup and initial activity as the previous. The base-runner on second, seeing the hit, chooses not to advance; batter-runner rounds first and heads to second; base-runner, seeing this, chooses to head toward third; fielding team throws the ball to third base.
At this point, the batter-runner realizes he made a mistake, leaves second base, and goes back to first. The base-runner then returns to second. For whatever reason, the fielding team doesn’t make a play. Both runners are safe (and the fielding team’s manager yells at everyone).
But now consider…
Same scenario, but the fielding team throws to first before the batter-runner returns.
As above. The batter-runner realizes he screwed up by pushing the base-runner off second when he couldn’t make it to third, and heads back toward first.
This time, the fielding team successfully throws the ball to the first baseman, who steps on the bag.
Does a force out apply? Is the batter-runner out, or is a tag required?
Yes, the batter-runner is forced out, and no, a tag is not required.
This is why baseball rules distinguish between the batter-runner and the base-runner, in order to create differentiation in these scenarios. Just because you’ve got guys running on the paths between bases, they are not equivalent runners. The batter-runner remains the batter-runner until the play is fully resolved, with all players either safe or out. Only at that point does the batter-runner, if he’s safe, become a base-runner.
In the above, by achieving a base hit, the batter-runner is forced to advance to first base. If he makes the turn and heads to second, but the base-runner there doesn’t move, the fact that the batter-runner is forced to first remains in effect. Hence, a throw to the bag achieves a force out, and a tag is not required to achieve an out.
Does this make sense?