#MATL , 21 bytes

"Kk@k=K@XK=>?4XKx}K&h

Try it online!. Or verify all test cases.

###Explanation

This processes each character in a loop. Each iteration compares the current character with the previous character. The latter is stored in clipboard K, which is initiallized to 4 by default.

The current character is compared with the previous one twice: first case-insensitively and then case sensitively. The current character should be deleted if and only if the first comparison was true and the second was false. Note that, since clipboard K initially contains 4, the first character will always be kept.

If the current character is deleted clipboard K should be reset (so the next character will be kept); otherwise it should be updated with the current character.

" % Take input string implicitly. For each char from this string:
 K % Push previous char, initiallized to number 4
 k % Convert to lower case. For numbers it rounds down
 @ % Push current char
 k % Convert to lower case. 
 = % True if current and previous chars are (case-insensitively) equal
 K % Push previous char
 @ % Push current char
 XK % Update clipboard K with current char. This doesn't affect the stack
 = % True if current and previous chars are (case-sensitively) equal
 >? % If first comparison was true and second was false
 4XKx % Reset clipboard K to 4
 } % Else
 K % Push previous char
 &h % Concatenate horizontally to gradually build the output string

MATL , 21 bytes

"Kk@k=K@XK=>?4XKx}K&h

Try it online!. Or verify all test cases.

Explanation

This processes each character in a loop. Each iteration compares the current character with the previous character. The latter is stored in clipboard K, which is initiallized to 4 by default.

The current character is compared with the previous one twice: first case-insensitively and then case sensitively. The current character should be deleted if and only if the first comparison was true and the second was false. Note that, since clipboard K initially contains 4, the first character will always be kept.

If the current character is deleted clipboard K should be reset (so the next character will be kept); otherwise it should be updated with the current character.

" % Take input string implicitly. For each char from this string:
 K % Push previous char, initiallized to number 4
 k % Convert to lower case. For numbers it rounds down
 @ % Push current char
 k % Convert to lower case. 
 = % True if current and previous chars are (case-insensitively) equal
 K % Push previous char
 @ % Push current char
 XK % Update clipboard K with current char. This doesn't affect the stack
 = % True if current and previous chars are (case-sensitively) equal
 >? % If first comparison was true and second was false
 4XKx % Reset clipboard K to 4
 } % Else
 K % Push previous char
 &h % Concatenate horizontally to gradually build the output string

#MATL, 21 bytes

"Kk@k=K@XK=>?4XKx}K&h

Try it online!. Or verify all test cases.

###Explanation

This proccesses each char in a loop. Each iteration compares the current character with the previous character. The latter is stored in clipboard K, which is initiallized to 4 by default.

The current char is compared with the previous one twice: first case-insensitively and then case sensitively. The current character should be deleted if and only if the first comparison was true and the second was false. Note that, since clipboard K initially contains 4, the first character will always be kept.

If the current character is deleted clipboard K should be reset (so the next character will be kept); otherwise it should be updated with the current character.

" % Take input string implicitly. For each char from this string:
 K % Push previous char, initiallized to number 4
 k % Convert to lower case. For numbers it rounds down
 @ % Push current char
 k % Convert to lower case. 
 = % True if current and previous chars are (case-insensitively) equal
 K % Push previous char
 @ % Push current char
 XK % Update clipboard K with current char. This doesn't affect the stack
 = % True if current and previous chars are (case-sensitively) equal
 >? % If first comparison was true and second was false
 4XKx % Reset clipboard K to 4
 } % Else
 K % Push previous char
 &h % Concatenate horizontally to gradually build the output string

#MATL, 21 bytes

"Kk@k=K@XK=>?4XKx}K&h

Try it online!. Or verify all test cases.

###Explanation

This processes each character in a loop. Each iteration compares the current character with the previous character. The latter is stored in clipboard K, which is initiallized to 4 by default.

The current character is compared with the previous one twice: first case-insensitively and then case sensitively. The current character should be deleted if and only if the first comparison was true and the second was false. Note that, since clipboard K initially contains 4, the first character will always be kept.

If the current character is deleted clipboard K should be reset (so the next character will be kept); otherwise it should be updated with the current character.

" % Take input string implicitly. For each char from this string:
 K % Push previous char, initiallized to number 4
 k % Convert to lower case. For numbers it rounds down
 @ % Push current char
 k % Convert to lower case. 
 = % True if current and previous chars are (case-insensitively) equal
 K % Push previous char
 @ % Push current char
 XK % Update clipboard K with current char. This doesn't affect the stack
 = % True if current and previous chars are (case-sensitively) equal
 >? % If first comparison was true and second was false
 4XKx % Reset clipboard K to 4
 } % Else
 K % Push previous char
 &h % Concatenate horizontally to gradually build the output string

#MATL, 21 bytes

"Kk@k=K@XK=>?4XKx}K&h

Try it online!. Or verify all test cases.

###Explanation

This proccesses each char in a loop. Each iteration compares the current character with the previous character. The latter is stored in clipboard K, which is initiallized to 4 by default.

The current char is compared with the previous one twice: first case-insensitively and then case sensitively. The current character should be deleted if and only if the first comparison was true and the second was false. Note that, since clipboard K initially contains 4, the first character will always be kept.

If the current character is deleted clipboard K should be reset (so the next character will be kept); otherwise it should be updated with the current character.

" % Take input string implicitly. For each character:
 K % Push previous character, initiallized to number 4
 k % Convert to lower case. For numbers it rounds down
 @ % Push current character
 k % Convert to lower case. 
 = % True if both characters are (case-insentively) equal
 K % Push previous character
 @ % Push current character
 XK % Update clipboard K with current character. This doesn't affect the stack
 = % True if both characters are (case-sensitively) equal
 >? % If first comparison was true and second was false
 4XKx % Reset clipboard K to 4
 } % Else
 K % Push previous character from clipboard
 &h % Concatenate horizontally to gradually build the output string

#MATL, 21 bytes

"Kk@k=K@XK=>?4XKx}K&h

Try it online!. Or verify all test cases.

###Explanation

This proccesses each char in a loop. Each iteration compares the current character with the previous character. The latter is stored in clipboard K, which is initiallized to 4 by default.

The current char is compared with the previous one twice: first case-insensitively and then case sensitively. The current character should be deleted if and only if the first comparison was true and the second was false. Note that, since clipboard K initially contains 4, the first character will always be kept.

If the current character is deleted clipboard K should be reset (so the next character will be kept); otherwise it should be updated with the current character.

" % Take input string implicitly. For each char from this string:
 K % Push previous char, initiallized to number 4
 k % Convert to lower case. For numbers it rounds down
 @ % Push current char
 k % Convert to lower case. 
 = % True if current and previous chars are (case-insensitively) equal
 K % Push previous char
 @ % Push current char
 XK % Update clipboard K with current char. This doesn't affect the stack
 = % True if current and previous chars are (case-sensitively) equal
 >? % If first comparison was true and second was false
 4XKx % Reset clipboard K to 4
 } % Else
 K % Push previous char
 &h % Concatenate horizontally to gradually build the output string