The problem on your code is the line that says

<?php if (is_archive()) { 

Category, Tag, Author and Date based pages are all types of Archives.

So on the videos category page, each post is showed by the block inside the

<?php if (is_archive()) { 

rather the

<?php } else { ?>

block as you would expect.

The easiest way to send the video category posts to the

<?php } else { ?>

block is creating a little filter. Change this if statement:

<?php if (is_archive()) { 

to something more specific:

<?php if (is_archive() && !is_category('video')) { 

This statement says: "If is archive but not video category then..."

This way your video category posts will end up on the

<?php } else { ?>

statement.