How to append to a file at certain times, and overwrite it at others?

In my program, I have a file which stores a list of names. I also have a way of deleting those names - writing the file contents to an vector, then erasing the name from the vector, then writing the contents of the vector back to the file. When a user wants to add a name, then the file appends it. My issue is that when I try to take out an entry, and replace the file with whatever is left, it appends that to the file.
So, for example, if one name is Admin, and another name is User, if a user tries to remove Admin, then the result looks like this:

-- Admin --

-- User --

-- User --

It retains Admin and User, and appends the result of erasing user.

This is my code:

	void create_password()
	{
		string holder;
		string keeper;
		vector<string> delete_entry;
		vector<string>::iterator new_end;
		
		ofstream New_PasswordFile(new_password_name + ".txt");//Creates a file called *name*.txt(used to store password for given name)
	ofstream Display_File;  		//Saves name to another file, which is diplayed to the user, when they attempt to retrieve a passwords

	//Display file is the file with all of the names
	
	Display_File.open(stored_username + "names.txt", ios_base::app);
	//Appends to the file instead of recreating it.
	ifstream Display_File_Checker(stored_username + "names.txt");
	//Opens same file under different name
	if(is_same_entry != true){//if a new entry is inputed
		Display_File << "\n-- " << new_password_name << " --\n"; //Inputs the new name into the file with -- *name* --
	}
	
	cout<<"WHAT IS THE PASSWORD?"; //Prompts to input password to save to the *name*
	cin >> new_password;
	
	if (new_password == "DELETE"){//if inputed password for given entry is delete
		new_password = "";

		while (getline (Display_File_Checker, holder))
		delete_entry.push_back(holder);//Adds display to vector delete_entry

for (int i = 0; i < delete_entry.size(); i++) { 
	if(delete_entry[i] == "-- " + new_password_name + " --"){
		cout << "working" << endl;
new_end  = remove(delete_entry.begin(), delete_entry.end(), "-- " + new_password_name + " --");

delete_entry.erase(new_end,delete_entry.end()); //erase the entry
	}
	}
	
	//write contents of new_end to entry file
	for (int i = 0; i < delete_entry.size(); i++){
		keeper = delete_entry[i];
cout <<  "keeper :" << keeper << endl; //<< endl;
		Display_File << keeper << endl; //appends instead of overwriting.
	}

	Display_File.close(); 
	}

I tried to not append to the file, but then users are unable to add entries. I'm not sure if there is a way to code this, or if there is a logical way to go about doing this.

Yes you are mixing several operations that should be apart.

If you want to update a file in place - that's what database are meant to do. IF you are considering that, sqlite3 would be a very good solution

However the typical solution here would be:

1. Load the entire file in memory, perhaps into a (hash) map

2. Make the operations you need - add/delete/modify

3. Write the entire map back to the file.