You can modify the comment form by passing arguments to the comment_form() function right in the theme. In Twenty Ten, that function is called in the comments.php file. You shouldn’t edit the wp-includes/comment-template.php file.
To just remove the allowed markup list from below the comment:
<?php comment_form('comment_notes_after='); ?>
To just remove the Website field:
<?php comment_form(array( 'fields' => apply_filters( 'comment_form_default_fields', array('url' => '')))); ?>
To change the name label to ‘Your Name’, change the email label to ‘Your Email’, change the comment label to ‘Let us know what you have to say:’, and remove the Website field and markup notes:
<?php $comment_args = array( 'fields' => apply_filters( 'comment_form_default_fields', array(
'author' => '<p class="comment-form-author">' .
'<label for="author">' . __( 'Your Name' ) . '</label> ' .
( $req ? '<span class="required">*</span>' : '' ) .
'<input id="author" name="author" type="text" value="' .
esc_attr( $commenter['comment_author'] ) . '" size="30"' . $aria_req . ' />' .
'</p><!-- #form-section-author .form-section -->',
'email' => '<p class="comment-form-email">' .
'<label for="email">' . __( 'Your Email' ) . '</label> ' .
( $req ? '<span class="required">*</span>' : '' ) .
'<input id="email" name="email" type="text" value="' . esc_attr( $commenter['comment_author_email'] ) . '" size="30"' . $aria_req . ' />' .
'</p><!-- #form-section-email .form-section -->',
'url' => '' ) ),
'comment_field' => '<p class="comment-form-comment">' .
'<label for="comment">' . __( 'Let us know what you have to say:' ) . '</label>' .
'<textarea id="comment" name="comment" cols="45" rows="8" aria-required="true"></textarea>' .
'</p><!-- #form-section-comment .form-section -->',
'comment_notes_after' => '',
);
comment_form($comment_args); ?>
To get the list above, I copied parts of the $defaults array from the documentation page for the comment_form() function:
https://codex.www.ads-software.com/Template_Tags/comment_form