Is there a way to keep this system but show the small image on the hom page without having in the post?
Not without custom programming or a third party plugin that changes the way the excerpt works in WordPress. Excerpts do not show any HTML, images included.
Any ideas why has an influence on what is shown on the home page?
Using the tag tells WordPress to use the_content() function, which displays HTML (including images), then it sets a cut off point.
Without the tag, the_excerpt() function is used, which doesn’t show any HTML.
You may have luck looking for a plugin which alters the functionality of the_excerpt() function to allow images, but I’m generally against tweaking that function as things can get buggy pretty quickly.