Unmerge ICloud account

Before signing out the the Apple ID on iPhone was notes , contacts box turned as on https://support.apple.com/guide/iphone/change-settings-iph7262fd4fe/ios

https://support.apple.com/en-in/HT205754

Set up iCloud Contacts

When you turn on iCloud Contacts, all of the information* stored on separate devices uploads to iCloud. If you turn off iCloud Contacts, your information won't automatically move from iCloud back to your device.

On your iPhone, iPad, or iPod touch

1. Go to Settings > [your name] > iCloud. 
2. Turn on Contacts.
3. When you're asked if you want to Merge or Cancel, tap Merge.

All of the information stored in Contacts uploads to iCloud.

After that you signed out this Apple ID https://support.apple.com/en-in/HT208242 in a single iPhone and then signed in with new Apple ID https://support.apple.com/en-in/HT204053

And then again signed out the new Apple ID and signed in with old Apple ID , if you ever had turned on contacts and notes box they would have been saved on iCloud server .

It seems you didn't turned as on that's the reason when you sign in with old Apple ID and password on the iPhone they were not on the iCloud server and zero notes / contacts data is shown .

if you have signed in with same Apple ID and password between two iPhones on the same network , the initial step is contacts / notes and all other data is backed up on iCloud server ( that secret is with you as you operate the device we don't know whether the boxes were checked or not ) .

Also as per this article https://support.apple.com/guide/iphone/back-up-iphone-iph3ecf67d29/ios

1. Go to Settings  > [your name] > iCloud > iCloud Backup.
2. Turn on iCloud Backup. iCloud automatically backs up your iPhone daily when iPhone is connected to power, locked, and on Wi-Fi.

You must remember that iCloud back up was turned as on , fortunately if it was then possibly the data can be restored ( including the notes / contacts .

See https://support.apple.com/en-in/HT204184

You signed in to the device with another ID?

What do you mean you merged your iCloud account with another’s? As far as I know, iCloud accounts cannot be merged.

As far as I know, one cannot merge iCloud accounts so I’m not even sure how you managed this.

That’s not merging accounts, that simply means that your bookmarks, reading lists, iCloud tabs will be synced across your devices.

Do make any changes elsewhere? That notification has nothing to do with Notes and Contacts.