What is the correct name?

This name may appear incorrect , but it is correctly applied. The reason is in order to show the configuration of the molecule ( think of cis, trans) . Following your naming proposal
Molecule1
..Cl ..... ...Br
....\........./
.....C = C
...../.......\
....F........I

And molecule 2:

..Cl ..... ...I
....\........./
.....C = C
...../.......\
....F........Br

Both would be - 2-bromo-1-chloro-1-fluoro-2-iodoethene . But they are clearly different molecules.
Unfortunately, many compounds cannot be described adequately by the cis-trans system. Consider, for example, configurational isomers of 1 -fluoro- 1 -chloro-2- bromo-2-iodo-ethene,shown above in molecule 1&2. There is no obvious way in which the cis-trans system can be used:
A system that is easy to use works as follows:
1. An order of precedence is established for the two atoms or groups attached to each end of the double bond according to the sequence rules which decides priority in terms of molar mass , not alphabetically. . When these rules are applied to 1-fluoro- 1-chloro-2-bromo-2- iodoethene, the priority sequence is:
• at carbon atom 1, Cl has priority over F
• at carbon atom 2, I has priority over Br
2. Examination of the two configurations shows that the two priority groups- one on each end- are either on the same side of the double bond or on opposite sides:
The Z isomer is designated as the isomer in which the top priority groups Cl and I, are on the same side .
The E isomer is designated as the isomer in which the top priority groups Cl and I, are on the opposite side
The two molecules would be named
E- 1 -fluoro- 1 -chloro-2- bromo-2-iodo-ethene, (opposite sides)
Z - 1 -fluoro- 1 -chloro-2- bromo-2-iodo-ethene, ( same side)
Note that the C1 substituted halogens are named together , and then the halogens on C2 are named together.

To name the groups in an organic compound, not only the groups are arranged in alphabetical order, but also the number set should be the smallest.

Hence, the correct IUPAC name is : 1-bromo-2-chloro-2-fluoro-1-iodoethene.

You're correct.