Most of the DHS surveys are based on a 2-stage design, in which sampling clusters are selected as PSU's in the first stage and households are selected in the second stage. To properly calculate the sampling variance, only the sampling clusters should be declared in the survey package. Therefore, I would say design (b) is the correct one

mydesign <- 
svydesign( 
id = ~V021, 
strata = ~V022 ,
data = data , 
weight = ~samplewt,
)

The sampling clusters can be id = ~V021 or id = ~V001. In most of the countries, both should be the same.

Regarding the fpc, for many reasons we do not provide the fpc value in the results report. Since we are dealing with national population surveys, the fpc value should be very small. Therefore, you can ignore the fpc in calculating the sampling errors and it should not affect the results.