When I used the method using the second screen shot of yours it didn't prompted my trusted mobile number but an Apple ID , while creating Apple ID initially , a rescue mail was created by me .
Followed these steps -
Shut down the Mac , press command and R and power key , the bars loads let it be completed , select language as English , click on encircled right arrow .
From the top menu bar > click on utilities > click on terminal > type resetpassword and click on enter key on the keyboard .
A window shows if you canât log into your computer because you forgot your password , you can reset it now .Select a user to reset password the password for , so click forcefully the user name and click on next .
A prompt will show enter the password for apple id and empty password field will be shown , donât type anything in the password field , click on forgot Apple ID and password ?
A small circle will rotate after a while a window will appear showing , enter your Apple ID to continue your Apple ID is the email address or phone number you use to sign in to the iCloud the App Store , or iTunes .The Apple ID will be written and two options will be shown ( a ) Back ( b ) continue so click on continue a small circle will rotate .
After a while , a window appears showing select how you want to reset your password .
( a ) Reset by email .
An email with reset instructions will be sent to your email address .
( b ) Answer security questions .
Answer security questions and change your password now .
You can check the box of reset by email .
Two options will be shown ( a ) back ( b ) continue . So click on continue .
A window appears password reset email sent .
an email has been sent to your rescue email address âŠ.Follow the directions in the email to reset your password .
Two options will be shown ( a ) back ( b ) ok , so click on ok .
Now , open your web mail rescue account by entering the mail address and password , in the mai inbox click on the mail it will be showing .
Dear âŠ.your first name and last name will be written .
You recently made a request to reset your id .Please click the link below to complete the process .
Reset now >
If you did not make this , if you did not make this change or you believe an unauthorised person has access your account , go to iforgot.apple.com to reset your password without delay .Following this , sign into your Apple ID account page at https://appleid.apple.com to review and update your security settings .
You have to click on reset now >
A window will appear , enter the new password , in the confirm password field type the same password , and click on reset password .
The Apple ID password is changed from recovery mode .
And a thought do you know how to change Apple ID password by opening system preferences > iCloud , and also using find my phone feature , resetting questions and answers for Apple ID in recovery mode , any thought will be highly appreciated or I can create a new thread and ask these questions , but you guided me in the right way,
Thanks .