Did you get these latest two budgies together from the same pet store?
If not then the latest arrival needs to be quarantined in a separate cage and room away from Rocky/Roxie and the budgie you got for your mother's birthday.
Quarantine IS Necessary!
The green budgie is a male while the blue one is a female. They are both fairly young, I would say the green budgie is slightly younger at about 4 and a half months old and the blue one is about 5 months old.
Can't your mother sit on a chair while having the cage on a table or stand and work on connecting and bonding with the budgie that way?
By having a small cage on the lap, this could stress the budgie out due to the very close proximity, not to mention that the cage will not be placed on a secure position and the chances for it to wobble are much higher.
During the training process, it's important to provide a safe, stable and stress free environment for the budgie.
Regarding the leg bands, they can be used to identify the breeder (by their initials) as well as the individual bird in case of an escape.
If a date is provided on the leg band, then a person can also tell the year the budgie has hatched.