Removing a payment method

Re: with my understanding the iCloud storage upgrade plan isn't considered a subscription?

It's not listed in my "Subscriptions" tab

Seems iCloud+ (anything more than the free 5Gb) is only available by subscription.

Eg: Apple says: "iCloud+ is Apple's premium cloud subscription".

in iCloud+ plans and pricing - Apple Support

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Re: "So, what I ultimately want to do is have JUST my Apple ID as a payment method

with nothing else, as well as avoid paying with my Apple Cash card."

Seems Apple must have found that did nor work, as your balance ran out one month ?

(If your iCloud stops being funded, you may lose synced data. For safe keeping, you may want to: Archive or make copies of the information you store in iCloud - Apple Support)

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Re: (1) "Do you think it'll let me remove the Apple Cash card after I pay? (Because of the prompt of "There is an active subscription that requires this payment method") Or is there a way to remove the Apple Cash card right now, before the payment?"

You can always try: However, seems Apple may require more than an Apple ID balance on file for some subscriptions, even if the Apple ID balance is used for the monthly payment.

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Re: (2) "Your default payment method is at the top".

In this case would Apple charge me with my Apple ID card, or Apple Cash card?

Generally, it seems Apple will always try to use the Apple ID Balance first, and then, if there's not enough to cover the payment, it then tries to use the other methods on file in order, from the top down as they are listed in your Apple ID account.

More info: How apps, content, and subscriptions from Apple are billed - Apple Support

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Re: (3) Is it possible to transfer money from the Apple ID card into Apple Cash?

It doesn't seem so ... an Apple ID balance can only be used for purchases from Apple.

See how to: Add money to your Apple Account balance - Apple Support

If needed:

You can: Transfer money in Apple Cash to your bank account or debit card - Apple Support

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Re: "(4) Are there any other methods of ensuring I pay from my Apple ID card,

like "assigning" a payment method to the iCloud upgrade plan?"

Anyone can forget to top up an Apple ID Balance ... seems Apple requires another payment method on file for some of its subscriptions. Your Apple ID Balance will be used as long as it has the money.

You can check which payment methods Apple can accept in your country or region. Payment methods that you can use with your Apple ID - Apple Support

If you add another "back-up" payment method, you can likely remove your Apple Cash Card.

How to: Add a payment method to your Apple ID - Apple Support

and

How to: Remove a payment method from your Apple ID - Apple Support

All the best :-)