Include all records in between two dates
So if you're trying to find all active/valid records in a time frame, and you are using intervals, then you must use the following criteria:

End date was on/after the start of the window
Start date was on/before the start of the window

This is working on the assumption that a record is still effective as of the end date, as indicated in your example.
So your query becomes:
SELECT
  Id
 ,start_time
 ,end_time
FROM
  &lt;your table&gt;
WHERE
  start_time &lt;= '2019-07-25'
    AND end_time &gt;= '2019-07-22'

A simple truth table helps verify:
Id  | Start_Time  | End_Time    | Start &lt;= 2019-07-25 | End &gt;= 2019-07-22
1   | 2019-07-22  | 2019-07-25  | Y                   | Y
2   | 2019-07-20  | 2019-07-25  | Y                   | Y
3   | 2019-07-20  | 2019-07-21  | Y                   | N
4   | 2019-07-20  | 2019-07-25  | Y                   | Y
5   | 2019-07-22  | 2019-07-22  | Y                   | Y
6   | 2019-07-25  | 2019-07-25  | Y                   | Y
7   | 2019-07-26  | 2019-07-27  | N                   | Y

Other notes:
Those are dates, not times.  Column should be Start_Date/End_Date or Start_Datetime/End_Datetime if you are include hours/minutes/seconds (you can abbreviate with Dt/Dtm if you wish).
From comment:

I assume that if I want to filter only for the startdate, I can do
WHERE end_time &gt;= '2019-07-22 00:00:00' But I am not sure about query
only for the enddate. What is the correct condition? WHERE start_time
&lt;= '2019-07-25 23:59:59' OR WHERE '2019-07-25 00:00:00' BETWEEN
start_time AND end_time

So here we need to be careful with what question is actually be asked, and what answer the query is providing.
The statement WHERE end_time &gt;= '2019-07-22 00:00:00' is the same as saying: any records that end OR start on/after 2019-07-22 00:00:00.
The inverse of that statement would be any records that start AND end before 2019-07-22 00:00:00, which is WHERE end_time &lt; '2019-07-22 00:00:00'.
So if your question is actually &quot;Which records end or start on/before 2019-07-22 00:00:00 the condition is WHERE start_time &lt;= 2019-07-22 00:00:00