What gets saved in a template
When you save your SharePoint site as a
template, you’re saving the overall framework of the site – its lists
and libraries, views and forms, and workflows. In addition to these
components, you can include the contents of the site in the template,
for example, the documents stored in the document libraries. This
could be useful to provide sample content for users to get started
with. Keep in mind that this could also increase the size of your
template beyond the default 50 MB site template limit.
Most of the objects in a site are included and supported by the
template. There are a number of objects and features not supported
however. The following table provides a quick summary of what’s in and
what’s out of a typical site template, or solution.
If you can make changes to your Master page then try this,
If you don’t want to edit every single WebPart page and/or if you want
to retain the ability to change the page title I have another solution
for you. My solution is to change the SiteLogoImage node name in the
master page. Thus the javascript generated by the PageTitleWebPart is
no longer able to change the logo.
Details
Open your master page and find the line starting with:
<SharePoint:SiteLogoImage name="onetidHeadbnnr0"
And replace the
“onetidHeadbnnr0” name with something else; “onetidHeadbnnr1” will do
the trick.
Reference
I believe you will need a new custom master page not new site template to show changes you made to a site's logo.

For More Info check links below:
Site Logo Not Changing on Web Part Pages
Introduction to SharePoint master pages