This is including all years, because the logic is easier to handle programmatically – if you have a reason that you wish some years removed, I’ll let you handle that aspect.

library(dplyr)
library(tidyr)
library(lubridate)
library(stringr)

df.a1 |> 
  # Set up date_from and date_to vars for period calculation
  mutate(date_from = if_else(str_detect(date, "from"), date, lag(date)),
         date_to = if_else(str_detect(date, "to"), date, lead(date))) |> 
  select(-date) |> 
  # Remove duplicates
  distinct() |> 
  # Further preparing for period calculation
  mutate(date_from = str_remove(date_from, "from "),
         date_to = str_remove(date_to, "to "),
         across(date_from:date_to, dmy),
         year_from = year(date_from),
         year_to = year(date_to)) |>
  # Setting data to long format in order to vectorize period calculation
  pivot_longer(year_from:year_to, names_to = "start_end", values_to = "year") |> 
  # Filling in all years and then resorting the rows back to original sort
  group_by(executive.name, district) |> 
  complete(year = full_seq(min(year):max(year), 1)) |> 
  ungroup() |> 
  arrange(district, year) |> 
  # Calculating the period for each year, district, and name
  mutate(period = case_when(year(date_from) == year(date_to) ~ date_to - date_from, 
                            start_end == "year_from" ~ as.Date(paste0(year, "-12-31")) - date_from,
                            start_end == "year_to" ~ date_to - as.Date(paste0(year, "-01-01")),
                            TRUE ~ as.Date(paste0(year, "-12-31")) - as.Date(paste0(year, "-01-01"))), .by = year) |> 
  # Remove unwanted names for a given district and year
  filter(period == max(period), .by = c(district, year)) |> 
  # Set data frame to desired output
  select(executive.name, year, district)

# A tibble: 20 × 3
   executive.name  year district
   <chr>          <dbl>    <int>
 1 Johnson         2000     1001
 2 Johnson         2001     1001
 3 Johnson         2002     1001
 4 Alleghany       2003     1001
 5 Clarke          2004     1001
 6 Clarke          2005     1001
 7 Roland          2006     1001
 8 Roland          2007     1001
 9 Richards        2008     1001
10 Richards        2009     1001
11 Peters          2000     1002
12 Peters          2001     1002
13 Peters          2002     1002
14 Harrison        2003     1002
15 Burr            2004     1002
16 Burr            2005     1002
17 Diamond         2006     1002
18 Diamond         2007     1002
19 Gorman          2008     1002
20 Gorman          2009     1002

df.a1 |>
  mutate(dt = dmy(date), year = year(dt), day = yday(dt),
         years = list(full_seq(year, 1)), 
         days = ifelse(str_detect(date, fixed("to")), day, 365 - day + 1),
         days = ifelse(n_distinct(year) == 1, max(dt) - min(dt), days),
         .by = executive.name) |>
  filter(max(dt) - min(dt) > 100, .by = executive.name) |>
  slice_max(days, n = 1, by = c(year, district)) |>
  mutate(years = map2(row_number(), list(pick(everything())), (rn, d) {
    setdiff(d$years[[rn]], d$year[d$executive.name != d$executive.name[[rn]]])
  }), .by = district) |>
  select(executive.name, years, district) |>
  distinct() |>
  unnest(years)
# # A tibble: 20 × 3
#    executive.name years district
#    <chr>          <dbl>    <int>
#  1 Johnson         2000     1001
#  2 Johnson         2001     1001
#  3 Johnson         2002     1001
#  4 Alleghany       2003     1001
#  5 Clarke          2004     1001
#  6 Clarke          2005     1001
#  7 Roland          2006     1001
#  8 Roland          2007     1001
#  9 Richards        2008     1001
# 10 Richards        2009     1001
# 11 Peters          2000     1002
# 12 Peters          2001     1002
# 13 Peters          2002     1002
# 14 Harrison        2003     1002
# 15 Burr            2004     1002
# 16 Burr            2005     1002
# 17 Diamond         2006     1002
# 18 Diamond         2007     1002
# 19 Gorman          2008     1002
# 20 Gorman          2009     1002

Some information on how this works:

1. mutate: this creates a date column dt and extracts date components for other computations (e.g. year, day of the year). A list-column years is created that contains all the years an executive.name served terms. As you mentioned some years have overlap, this is take care of later. Also, days column is created. This contains the number of days an executive.name served within a year. This is used as a tie breaker for those that served part of a year. The year is assigned to the executive.name that served the largest portion of the year in a later step. A small correction is made to days to ensure the right output if they served less than a year.
2. filter: this keeps rows that served more than 100 days. This excludes “Tollson”. You can set this filter to be different based on your needs.
3. slice_max: this removes rows within a county and year when there was more than one executive.name. By taking the max days, we keep the row of the executive.name that served a higher proportion of the year within a district.
4. mutate: this step is a bit tricky, but years is for every year an executive.name served a term. However, in split years you want to assign that district and year to one individual that served the most days. This step essentially removes those years from a term span that were not assigned to that individual. For example, Roland started in 2005 in district 1001. Since Clarke served a larger portion of that year, that year is assigned to Clarke, so we remove it from Roland’s years span.
5. The last three steps (select, distinct, and unnest) simply format the data into a long format.