I'm assuming you want 2008 to be 2008 and all the other years to have just the later values so I would use case_when. This is only works if the date is an integer as in your reprex:
library(dplyr)
data <- toy_data %>%
mutate(extractedYear = case_when(year == 2008 ~ 2008,
TRUE ~ year - 2000))
If the year in your actual data is in a date format you will need to add the following line of code and you may have to change the equation as the year may come through as something like 20200101:
library(dplyr)
data <- toy_data %>%
mutate(Year = as.integer(Year),
extractedYear = case_when(year == 2008 ~ 2008,
TRUE ~ year - 2000))