How to find the counts of word occurrences in a column?
You need to first split each string into multiple words and count the number of times each word occurs inside the string. Splitting can be done using regexp_split_to_table() which yields one row per word:
select w.word, w.num_occurrences
from the_table t
  cross join lateral (
     select word, count(*) as num_occurrences
     from regexp_split_to_table(lower(t.text_col), '[\s[:punct:]]+') as x(word)
     where word &lt;&gt; ''
     group by word
  ) w

This returns the following given your sample data:
word | num_occurrences
-----+----------------
test |               1
a    |               1
is   |               1
this |               1
test |               2
a    |               2
is   |               1
test |               1
how  |               1
to   |               1
test |               1
more |               1
one  |               1

This can be grouped by num_occurences to get the result you want:
select count(*), num_occurrences 
from (
  select w.word, w.num_occurrences
  from the_table t
    cross join lateral (
       select word, count(*) as num_occurrences
       from regexp_split_to_table(lower(t.text_col), '[\s[:punct:]]+') as x(word)
       where word &lt;&gt; ''
       group by word
    ) w
  where w.word = 'test'
) t
group by num_occurrences
order by 1 desc

And the result of that is:
count | num_occurrences
------+----------------
    3 |               1
    1 |               2