Is it ok to mention the original source and author but still claim copyright on the file as in the following header?
Should I mention the use of the MIT code elsewhere, for example in a license.txt file?
Reading the common form of the MIT license again and with the help of the answer of gnasher729, I answered most of my questions. The new code added in the file is copyrighted by me and the part that hasn't been modified is still copyrighted by the original author. I don't need to mention somewhere else the use of the MIT code.
However, the copyright notice below and the permission notice shall be included in all copies or substantial portions of the Software. This indicates that I should copy the MIT license and the name of the author above every portion of the code that I haven't modified. This is pretty much infeasible, so my guess is that mentioning the original author with a link to the original code is enough. I should also add the MIT license copyright notice in the file as well but this will add a overhead to every client downloading the website, so I probably won't.
Copyright (c) <year> <copyright holders> Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.