That was 70 years ago. Different period with different with different technologies demand different ways of doing things. And while both did use battleships on occasions, Nimitz seems to have spent more time with his flag on light cruisers and Halsey on carriers than battleships.
Makes sense really. This was a period before reliable high baud long range communications and automatic encryption systems, communications was laborious and slow. If you wanted detailed information you often had to go see and talk with the people responsible yourself. Command in a sea battle between ships involved watching the enemy out a window, manual tracking plots were usually behind and often in error. In an air battle the best information was in the radio room of one of the carriers listening to the reports as they came in, so that is where most of the commanders would be.
Nimitz was the Commander-in-Commander Pacific Ocean Area, not a fleet commander, which covering the vast area from California to Japan and from the Aleutians to Australia. So Nimitz was almost continuously on the move. It is a little difficult to justify detaching a battleship as the CinC’s private yacht or even a heavy cruiser as these are major combatants, but a light cruiser is not, and they were among the fastest vessels in the fleet.
Halsey commanded the Southern Pacific Ocean Area, 1/3 of Nimitz’s command, which allowed him to mostly travel with the fleet. He preferred carriers because that was where the action was, with their greater communications ability and immediate access to all aircraft reports. Carriers epitomized his slogan of “Hit hard, hit fast, hit often”, and he wanted to be there to do it, to the point of seeing frequently the older (slow) battleships as more of a hindrance than an asset to his fast carriers.