BEM. Block and its modifier?

Question

css

There are several different headers on the site, would it be correct to make a .header block with modifiers that change it to the desired style.

<div class="page">
  <div class="page__header">
    <h1 class="header header_page">Заголовк страницы</h1>
  </div>
  <div class="page__slider">
    <div class="slider">
          <div class=slider__item> 
            <h1 class="header header_slider">Заголовк слайдера</h1>
          </div>
    </div>
  </div>
<div>

.header{}
.header_slider{}
.header_page{}

Roughly speaking, .header will be empty and all styles will be in .header_slider and .header_page
.

Answer 1

Or maybe you should not give up the context and start working normally already?)
So much time is spent on comprehending the zen of BEM...