How to write the label into the plot in ggplot2

I have a plot with the labels on the y axis (after using coord_flip)
Here is a reproducible example

Names <- c("ExampleA","ExampleB","ExampleC","ExampleD", "ExampleE", "ExampleF","ExampleG", "ExampleH")
Counts <- c(4,3,2,1,-1,-2,-3,-4)
Type <- c("X","X","X","X", "Y","Y","Y","Y")
df <-data.frame(Names,Counts, Type)

ggplot(df, aes(x=reorder(Names,Counts), y=Counts, fill=Type))+
  geom_col()+
  coord_flip()+
  theme_minimal()

I want the labels to be written next to the bars. Meaning that "ExampleA/B/C/D" etc. should be written right next to the bars and not listed on the y axis.
Or in visuals:
What I have

What I want

(Credit Li et al. 2021 https://doi.org/10.3389/fimmu.2021.694801)

I tried:
https://stackoverflow.com/questions/55406829/ggplot-put-axis-text-inside-plot
which didn't work.

This

geom_label(label=Names, parse=TRUE)

is close but not really what I want (and also for the real data only works when I type out every single name into a vector which I'd like to avoid if not nescessary).
I was hoping that there is something along the lines of

geom_label(label=Names, position="inside")

or

scale_x_discrete(position = "inside")

would be possible, but thus far I could not find anything.

Updated to match OP's required output and long names.

Have rearranged x and y subjects from original so that we can avoid coord_flip and read the graph as is, rather than transpose x and y.

Made text positioning more robust using both hjust and nudge_x, the former aligns the text and the later provides the padding between the text and the end of the bar.

As for the long text this is managed by creating extra space either side of the plot using expansion in the call to scale_x_continuous and setting the text size to 3 in the call to geom_text. An alternative would be to edit the text to include a line break but this may be problematic where there are many bars.

library(ggplot2)

Names <- c("ribonucleoside triphosphate biosynthetic process" ,"purine ribonucleoside triphosphate biosynthetic process" ,"ATP biosynthetic process" ,"proton motive force-driven ATP synthesis" ,"NADH dehydrogenase complex assembly" ,"mitochondrial respiratory chain complex I assembly" ,"innate immune response" ,"antigen processing and presentation of peptide antigen")
Counts <- c(4,3,2,1,-1,-2,-3,-4)
Type <- c("X","X","X","X", "Y","Y","Y","Y")

df <-data.frame(Names,Counts, Type)

ggplot(df, aes(Counts, reorder(Names,Counts), fill=Type))+
  geom_col()+
  geom_text(aes(label = Names, x = 0),
            size = 3,
            nudge_x = ifelse(Counts < 0, 0.05, -0.05),
            hjust = ifelse(Counts < 0, 0, 1)) +
  scale_x_continuous(expand = expansion(mult = c(0.75, 0.75)))+
  theme_minimal() +
  theme(axis.text.y = element_blank(),
        legend.position = "bottom")

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^{Created on 2023-06-26 with reprex v2.0.2}

Alternatively, you could create new variables x2 and y2 as below and use them in annotate function to generate the expected plot

df <-data.frame(Names,Counts, Type) %>% arrange(desc(Type), desc(Names), Counts) %>% 
mutate(x2=row_number(),
       y2=ifelse(Type=='X',-0.4,0.4)) %>% 
  arrange(Type,Names,Counts)

ggplot(df, aes(x=reorder(Names,Counts), y=Counts, fill=Type))+
  geom_col()+
  coord_flip()+
  theme_minimal() + annotate(geom = 'text', y=df$y2, x=df$x2, label=df$Names) + 
  theme(axis.text.y = element_blank())

# output

      Names Counts Type x2   y2
1 ExampleA      4    X  8 -0.4
2 ExampleB      3    X  7 -0.4
3 ExampleC      2    X  6 -0.4
4 ExampleD      1    X  5 -0.4
5 ExampleE     -1    Y  4  0.4
6 ExampleF     -2    Y  3  0.4
7 ExampleG     -3    Y  2  0.4
8 ExampleH     -4    Y  1  0.4

If the size of the Names is lengthy then we can decrease the size of the labels in annotate() as below

ggplot(df, aes(x=reorder(Names,Counts), y=Counts, fill=Type))+
  geom_col()+
  coord_flip()+
  theme_minimal() + annotate(geom = 'text', y=df$y2, x=df$x2, label=df$Names, size=3) + 
  theme(axis.text.y = element_blank())