I added the following code sequence in my custom theme's default.xml
file:
<container name="header.main.search" htmlTag="div" htmlClass="header-search" after="-">
<block class="Magento\Framework\View\Element\Template" template="Magento_Search::form.mini.phtml" />
</container>
<move element="header.main.search" destination="header-wrapper" after="logo" />
<move element="top.links" destination="header-wrapper" after="header.main.search" />
The code above renders the following HTML in the page source:
<div class="header content">
<strong class="logo">
<img src="https://dev3.pro-detailing.ro/pub/static/version1516374892/frontend/ProDetailing/v1/en_US/images/logo.png" alt="Magento Commerce" width="263" height="85">
</strong>
<div class="header-search">
<form class="navbar-form navbar-right" role="search" id="search_mini_form" action="https://dev3.pro-detailing.ro/catalogsearch/result/" method="get">
<div class="form-group">
<input type="text" name="q" value="" placeholder="Search entire store here..." class="form-control" maxlength="128" role="combobox" aria-haspopup="false" aria-autocomplete="both" autocomplete="off">
</div>
<button type="submit" class="btn btn-default" disabled="">Search</button>
<div id="search_autocomplete" class="search-autocomplete" style="display: none;"></div>
</form>
</div>
<ul class="list-inline clearfix header links">
<li><a href="https://dev3.pro-detailing.ro/customer/account/">My Account</a></li>
<li class="authorization-link" data-label="or">
<a href="https://dev3.pro-detailing.ro/customer/account/login/">
Sign In</a>
</li>
</ul>
</div>
This is what I was looking for.