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> Do you reckon it can make a difference?
It doesn't matter. I use absolute URL's for image paths in include pages for multiple reasons, none of them having to do with optimization for the SE's. It has to do with optimizing for connectivity when the includes are being referenced from various resources.
> his website name contains the keyword.
This may be the reason why. There have been many discussions on having the keyword in the domain and this may be further proof that it is factored into the algo.
What you are referring to is called a canonical and there have been many discussions on this strategy also. Use the Site Search feature at top and search for "canonicals". There are a few who post here who can provide expert advice on this one. I've never researched or tested the strategy.
My gut instinct tells me that the keyword in the domain name (keyword.com) will always outweigh the keyword in a canonical (keyword.domainname.com) if all other factors are equal.