Thanks @Andreas_Hegenberg to improve this feature!!
I have installed the version 6.352 with very good results. I tell you the different behaviours I'm getting depending on different actions configurations:
CASE 1: Action "Run Applescript (async in background)" => "Source Type: Apple Script From SCPT File"
a) If I click the "Run Script" button inside BTT action, the four notifications are shown 
b) If I trigger the keyboard shortcut with BTT editor opened, only the 2nd notification is shown 
c) If I trigger the keyboard shortcut with BTT editor closed, only the 2nd notification is shown 
CASE 2: Action "Run Applescript (blocking)" => "Source Type: Apple Script From SCPT File"
a) If I click the "Run Script" button inside BTT action, only the 2nd notification is shown 
b) If I trigger the keyboard shortcut with BTT editor opened, only the 2nd notification is shown 
c) If I trigger the keyboard shortcut with BTT editor closed, only the 2nd notification is shown 
CASE 3: Action "Run Applescript (async in background)" => "Source Type: Apple Script" => "Inline Text"
a) If I click the "Run Script" button inside BTT action, the four notifications are shown 
b) If I trigger the keyboard shortcut with BTT editor opened, the four notifications are shown 
c) If I trigger the keyboard shortcut with BTT editor closed, the four notifications are shown 
CASE 4: Action "Run Applescript (async in background)" => "Source Type: Apple Script" => "External File (Choose Existing)"
a) If I click the "Run Script" button inside BTT action, the four notifications are shown 
b) If I trigger the keyboard shortcut with BTT editor opened, the four notifications are shown 
c) If I trigger the keyboard shortcut with BTT editor closed, the four notifications are shown 
CASE 5: Action "Run Applescript (blocking)" => "Source Type: Apple Script" => "Inline Text"
a) If I click the "Run Script" button inside BTT action, the four notifications are shown 
b) If I trigger the keyboard shortcut with BTT editor opened, the four notifications are shown 
c) If I trigger the keyboard shortcut with BTT editor closed, the four notifications are shown 
CASE 6: Action "Run Applescript (blocking)" => "Source Type: Apple Script" => "External File (Choose Existing)"
a) If I click the "Run Script" button inside BTT action, the four notifications are shown 
b) If I trigger the keyboard shortcut with BTT editor opened, the four notifications are shown 
c) If I trigger the keyboard shortcut with BTT editor closed, the four notifications are shown 
Case 1 and 2 are weird because "Run Script" button is working as expected when the applescript is running in background, but the rest of options in both cases are not showing the four notifications.
Anyway now the notifications are working much better! 