Can someone give an example of software that is distributed as both service and product and how to classify, if i have to eventually, under one category?
Can someone give an example of software that is distributed as both service and product
An example that comes quickly to mind is Wordpress. It's distributed under GPLv2+, so you can take a copy and install it on your own website, as very many people do. Most major distros repackage it, to make doing your own installation very easy (CentOS 7:
yum install epel-release ; yum install wordpress). However, you can also buy Wordpress site hosting as a service, from many providers (example 1, example 2, example 3, and I should stress these are taken from the results of a duckduckgo search and their selection implies no endorsement whatsoever).
how to classify, if i have to eventually, under one category
I'm sorry, I don't understand that question. If you mean "how do I decide whether I'm distributing software or offering SaaS", the question to ask yourself is "am I giving copies of software to people to run on their own computers, or am I running it on my own computer and letting them use it?".