SECTION 30 INFIELD FLY RULE
Infield fly rule is, when declared by the umpire, a fair fly (not including a line drive or an attempted bunt) that can be caught by an infielder with ordinary effort when runners are on first and second or all three bases are occupied and before there are two outs in the inning. Any defensive player positioned in the infield at the time of the pitch shall be considered an infielder for the purposes of this rule. The rule does not preclude outfielders from being permitted to make the catch. The ball is live, the batter is out, which removes the force, but runners may advance at their own risk. The runners may tag up and advance as soon as the batted ball is touched by a fielder. If a declared infield fly becomes foul, it is treated as a foul ball, not an infield fly.

In the second to the last sentence "The runners may tag up and advance as soon as the batted ball is touched by a fielder." Does this mean that on a called IFF, whether the fly is dropped or caught, on the touch the batter is out and runners do not need to tag up at all? TIA

The batter is out, period. Touching the ball has nothing to do with that part of the play.
Just like any other fly ball, BR are at risk and must tag up.

'In the second to the last sentence "The runners may tag up and advance as soon as the batted ball is touched by a fielder." Does this mean that on a called IFF, whether the fly is dropped or caught, on the touch the batter is out and runners do not need to tag up at all? TIA'

1..The batter is out when the IFF is called, not when the ball is touched.

2..As in ANY flyball, the runners MUST tag up, and are released as soon as the ball is touched.

Bob

Agree that IFF batter is out when called correctly
Runner can run on the hit and if not caught they are ok if they reach the base safely .
If caught however they must return to tag up or risk an appeal for leaving early

Should read: As in any CAUGHT fly ball, the runner's are subject to being ruled out on proper appeal should they leave the base prior to first contact by a member of the defense or not retouch the base after first contact.