Hello,
As much as I have understood your requirement, combination logo (minimalist icon+name) will be suitable for you . You can use that icon as your fav icon, different social media profiles and entire logo will use in the website.
I am an expert and 1+ year of experienced Graphic Designer like to work on your PetSupacentre logo designing project. I will use Adobe Illustrator for designing your logo. I already have designed logo for E Commerce portal, online jewelry selling site, software firm, online karate training portal, mining firm etc.
According to your requirement, I will show you 2 concepts first. If you like one of them, then ok, if you do not like then I would like to request you to be more specific regarding the requirement and will share more 2/3 concepts on that. Please do not worry because you will get scopes of modifications.
You can check my Freelancer portfolio and for more details. Please ping me over my Freelancer personal chat box where I can share you my recent works there.
Thanks a lot
Suparna