{% extends "wagtailadmin/shared/field.html" %}
{% load i18n wagtailadmin_tags %}

{% block form_field %}
    <div class="w-grid w-auto-cols-auto w-gap-4">
        {{ field }}

        <button type="button" id="{{ field.auto_id }}-chooser" class="button button-secondary">{% trans "Choose from popular search terms" %}</button>
    </div>
{% endblock %}

{% extends "wagtailadmin/shared/field.html" %}
{% load i18n wagtailadmin_tags %}

{% block form_field %}
    {% field_row %}
        {{ field }}

        <button type="button" id="{{ field.auto_id }}-chooser" class="button button-secondary">{% trans "Choose from popular search terms" %}</button>
    {% endfield_row %}
{% endblock %}

Use field_row template tag for search term picker field
Use field_row template tag for search term picker field
