If a child begins at another school, their place is relinquished
That is true if the child begins at another state school. The situation is less clear cut if the parents defer entry to the state school and the child enters an independent school. In this particular case the parents have not informed the state school that their child is at an independent school so there is currently no need for them to try and work out how the Admissions Code and the Pupil Registration Regulations interact in this situation. As Rafa points out, if the independent school place is a nursery place the state school place is definitely not relinquished.
If a child starts school somewhere else, they will get a UPN (unique pupil number). This will show their start date of school and nullify the request to delay start.
Not necessarily true where the pupil starts at an independent school as in this case. Independent schools are not required to use the UPN system.
Also, a delayed start is not a given and, at this moment, a school does not have to agree
Rubbish. The Admissions Code is clear. Parents can defer starting their child until the start of term following the child's fifth birthday or the start of the summer term whichever is earlier. The school has no choice. They cannot refuse a request to defer entry.
but I thought the place had to be held until beginning of year 1 for summer borns
No. For summer born children the place has to be held until the start of the summer term.