sizeof() do not count terminator”

sizeof reports the size of the allocated memory. There is no counting involved. You might be mistaking sizeof for strlen, which indeed does not count the string terminator.

The first one is auto-allocated, and is size 11 because you have 10 characters and NUL. The second is manually allocated to size 11, which is filled with 11 characters, with no NUL since there is no space for it; if you try to print it (or indeed apply strlen on it), you will get gibberish at best.

See Why does gcc allow char array initialization with string literal larger than array?

The sizeof operator returns the number bytes needed to store a variable (or a type). For string0 the compiler deduces that you have sizeof "0123456789" which is indeed 11 byte string literal including the trailing ''.

char string1[11] tells the compiler to allocate 11 bytes. As you initialize the variable to a 12 byte string literal only the first 11 bytes are copied to the variable so it’s actually an array of bytes (without the trailing '') and not a string.

Here is what C 2023 6.7.10 §10 says:

An array of character type may be initialized by a character string literal or UTF-8 string literal,
optionally enclosed in braces. Successive bytes of the string literal (including the terminating null
character if there is room or if the array is of unknown size) initialize the elements of the array.

OK, than you all, now is clear.
I post some code to show this behave:

#include <stdio.h>

char  string0[];                 // space for '', initialized
char  string1[10];               // space for 10 Bytes, initialized
char  string2[10]="";            // space for 10 Bytes, initialized
char  string3[]="";              // space for '', initialized
char  string4[]=  "0123456789";  // space for 11 Bytes, 10 plus ''
char  string5[11]="0123456789A"; // space for 11 Bytes, 11 char, miss ''
char  string6[10]="0123456789";  // space for 10 Bytes, 10 char, miss ''
char* string7;                   // space for sizeof(ptr), not initialized
char* string8=NULL;              // space for sizeof(ptr), initialized to NULL
char* string9=string4;           // space for sizeof(ptr), initialized

//stringX; // same as &stringX[0]

int main() {
   printf("size:%02ld string0:'0123456789A'n", sizeof("0123456789A")); // 12 '0123456789A'
   printf("size:%02ld string1:'%s'n", sizeof(string1), string1);       // 10 ''
   printf("size:%02ld string2:'%s'n", sizeof(string2), string2);       // 10 ''
   printf("size:%02ld string3:'%s'n", sizeof(string3), string3);       //  1 ''
   printf("size:%02ld string4:'%s'n", sizeof(string4), string4);       // 11 '0123456789'
   printf("size:%02ld string5:'%s'n", sizeof(string5), string5);       // 11 '0123456789A'
   printf("size:%02ld string6:'%s'n", sizeof(string6), string6);       // 10 '0123456789'
   printf("size:%02ld string7:'%s'n", sizeof(string7), string7);       //  8 NULL
   printf("size:%02ld string8:'%s'n", sizeof(string8), string8);       //  8 NULL
   printf("size:%02ld string9:'%s'n", sizeof(string9), string9);       //  8 '0123456789'
   
   printf("string4[9] : %02d '%c'n", string4[9] , string4[9]);  // '9'
   printf("string4[10]: %02d '%c'n", string4[10], string4[10]); // ''
   
   printf("string5[9] : %02d '%c'n", string5[9] , string5[9]);  // '9'
   printf("string5[10]: %02d '%c'n", string5[10], string5[10]); // 'A'
   printf("string5[11]: %02d '%c'n", *(string5+11), *(string5+11)); // '' confirm NULL
   printf("string5[12]: %02d '%c'n", *(string5+12), *(string5+12)); // ''

   printf("string6[9] : %02d '%c'n", string6[9] , string6[9]);  // '9'
   printf("string6[10]: %02d '%c'n", *(string6+10), *(string6+10)); // '' confirm NULL
   printf("string6[11]: %02d '%c'n", *(string6+11), *(string6+11)); // ''

   printf("string4 addr:%pn", string4);
   printf("string5 addr:%pn", string5);
   printf("string6 addr:%pn", string6);
   return 0;
}