A simple way in awk is just keep a flag and set n = 1 when "Start" is encountered in field 4. Checking n == 1 gives you a test to print each line while n is 1. When "End" is reached, simply print that line and exit, e.g.
awk '$4 == "End" {print; exit} $4 == "Start" {n = 1} n == 1' file
(note: the operation associated with the rule n == 1 is simply the default operation of print. It is equivalent to writing n == 1 {print})
Example Use/Output
Using your data file in file, you would get:
$ awk '$4 == "End" {print; exit} $4 == "Start" {n = 1} n == 1' file
2019/10/17 23:01:02 -I- Start
2019/10/17 23:01:02 -I- ======================
2019/10/17 23:05:02 -I- Summary Details
2019/10/17 23:10:02 -I- End
It can also be written as:
awk '$4 == "Start" {n = 1} n == 1; $4 == "End" {exit}' file
Edit - Todays Date Only
If you want to match today's date only, you simply need to pass the date in the format you need as a variable to awk from the shell using the -v option, e.g.
awk -v date="$(date +%Y/%m/%d)" '
    $1 == date { 
        if($4 == "Start") n = 1
        if(n == 1) {print}
        if($4 == "End") exit
    }
' file
Example Input
$ cat file
2019/10/17 23:01:02 -W- Blah Blah Blah.....
2019/10/17 23:01:02 -I- Blah Blah Blah.....
2019/10/17 23:01:02 -I- Blah Blah Blah.....
2019/10/17 23:01:02 -I- Start
2019/10/17 23:01:02 -I- ======================
2019/10/17 23:05:02 -I- Summary Details
2019/10/17 23:10:02 -I- End
2019/10/17 23:10:02 -I- Blah Blah Blah.....
2019/10/17 23:10:02 -I- Blah Blah Blah.....
2019/10/17 23:10:02 -I- Blah Blah Blah.....
2019/10/20 23:01:02 -W- Blah Blah Blah.....
2019/10/20 23:01:02 -I- Blah Blah Blah.....
2019/10/20 23:01:02 -I- Blah Blah Blah.....
2019/10/20 23:01:02 -I- Start
2019/10/20 23:01:02 -I- ======================
2019/10/20 23:05:02 -I- Summary Details
2019/10/20 23:10:02 -I- End
2019/10/20 23:10:02 -I- Blah Blah Blah.....
2019/10/20 23:10:02 -I- Blah Blah Blah.....
2019/10/20 23:10:02 -I- Blah Blah Blah.....
Example Use/Output (for 2019/10/20)
$ awk -v date="$(date +%Y/%m/%d)" '
>     $1 == date {
>         if($4 == "Start") n = 1
>         if(n == 1) {print}
>         if($4 == "End") exit
>     }
> ' file
2019/10/20 23:01:02 -I- Start
2019/10/20 23:01:02 -I- ======================
2019/10/20 23:05:02 -I- Summary Details
2019/10/20 23:10:02 -I- End
Let me know if you have further questions.