Family Sharing and Apple ID Balance

Yes, that is the case.

Confirming the Organizer's payment method in no way means it will use it for the purchase. It just means it needs to be confirmed before it can move forward.

Once confirmed, it will use first and foremost any available account balance on your account, before using the Organizer's payment method.

Yes, your family members can see your purchase in your purchase list, unless you hide them.

click here ➜ Hide purchases from the App Store - Apple Support

To address your last question first:

Hide App Store purchases -- Hide purchases from the App Store - Apple Support --> "If you use Family Sharing and hide apps, the apps will also be hidden from your family members."

Hide and unhide music, movies, TV shows, audiobooks, and books - Hide and unhide music, movies, TV shows, audiobooks, and books - Apple Support - "If you use Family Sharing, family members won't see items that you hide, unless they downloaded the item before you hid it." Hiding them means they will not show in your available downloads list but they will still appear in your purchase history. Music, movies and TV shows have to be hidden on a computer with the Music or iTunes apps. Read the document for full details.

As for the payment, you are correct, your account balance will be used first if it can be used for the particular items you are purchasing. However, the organizer must have a functioning payment method even if you will end up using your money in the end. Check with them that their method is working. Have the organizer read this document about how to view current payment types and add Apple ID payment methods on an Apple mobile device, a Mac, or a PC --> Add a payment method to your Apple ID - Apple Support

If an item appears in red then it is no longer working and it either needs to be replaced or made working again.

If you are not the organizer you cannot have any payment method other than personal balance. If there's a problem with the method it is the organizer's method that has the problem and the organizer needs to resolve it.

Yes, hidden purchases will not appear to others. However, whoever ends up paying for it will receive a receipt for the purchase and may then know about it. It won't, however, appear in the free downloads when others try to download it.

You cannot modify the Organizer's payment method or update it in any way. It just has to be verified. This will not notify the Organizer.

Any Payment methods you add to your account will not be used.

Even if the organizer's payment method gets charged, the organizer will not know what it's for, as it will only appear as "apple.com/bill" on their monthly statement and would only have the full name and description in your own Purchase history.

Secondly, if I were to update the payment method, would the organizer be notified? And to clarify once again, any purchase I make, assuming it to be hidden, would not be seen by anyone in the family nor the organizer?

Correct. Nobody can see any purchases you make, if they are hidden. You can always turn off Purchase Sharing on your side, so nothing you buy will ever be visible to anybody else in the family.