How to counteract the stage 5 boss spinning?

Your intuition is correct, though the strategy changes on the Hard difficulty.

On Easy and Medium, it is sufficient to just hold down L and R to counteract the rotating:

Per the Gamespot Walkthrough:

There are some pits that will warp you to different parts of the stage and some will spin you around. Use L or R to counteract the spinning.

Similarly, per the Strategy Wiki walkthrough:

Scorpellum: This thing is one of the hardest bosses in the game. The circular area around the boss will rotate like the spinners on the stage. Press L button or R button to stop the spinning.

On Hard, however, the rotations will be too fast to just hold down L or R. Instead, you need to DOUBLE TAP the L and R buttons. Per the IGN Walkthrough:

The spinning here is very strong and disorienting. DOUBLE TAP to stop your extreme clockwise movement and adjust your aim.

This is admittedly difficult to do (you're playing Contra on Hard, so it should be pretty challenging), though as you can see in the videos you linked it is possible to master this technique.

Some additional advice for beating this boss on Hard, again from the IGN Walkthrough:

Line yourself up with the eye and dodge the fire and worms when they threaten. It sucks when you have the eye in a dead aim and then have to duck a fire but it is better to lose your shot than to die. Stay to the outside as the worms are less numerous and the fire is easier to anticipate.

Also some advice for beating this boss on co-op:

It helps to have two people as you now have more firepower aiming at the boss and the worm things at the beginning will sometimes be distracted by the other player. No real teamwork is needed but beware that sometimes a worm will go after your partner and he'll either run the worm into you or the worm will switch targets. Make sure you circle the eye in the same direction: clockwise or counterclockwise.