You wrote
  <section>
   <item>I am item number 1</item>
   <item>I am item number 2</item>
   <item>I am item number 3</item>
  
  
  
  <xsl:template match="section">
   <xsl:apply-templates select="following-sibling::item[position() = 3]"/>

you are sat at a section node, and you have asked to go from there
to the third item node that is a sibling of the current node.
That isn't what you want as the item nodes are children of section, and
you want all items after position 2, not just the third.

<xsl:template match="section">
 <xsl:apply-templates select="item[position() &gt; 2]"/>


David


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