Tag documents with DocumentType field inside different sites

I have this question for our SharePoint sites which we are currently creating. Currently we need to create sites for our departments (10++), will take Commercial & HR as an example :-

1) HR

2) Commercial

Now inside the 2 sites we need to have a field named "Document Type" of type managed metadata to tag documents inside the documents libraries that got uploaded inside the 2 sites.

Now the issue is that the HR has different options compared to the Commercial site for the Document type field (some options might be the same such as the "Other" option) . so i am planning to follow this approach:-

1) Inside the HR site to create a managed metadata column with internal name = "DocumentType", and link it to this term set named "HR Document Type":-

2) Inside the Commercial site to create a managed metadata column with the same internal name = "DocumentType", and link it to this term set named "Commercial Document Type":-

now this will work on paper for tagging documents with different options for each site. but we need to have an advance search page to allow to filter the documents from the 2 sites using the Document Type, for this i will use the PnP Modern Search web part, which depend on the search managed metadata & the search Refinables.

So now i got one managed metadata for the DocumentType columns:-

and i linked it to a RefillableString as follow:-

then i am planning to configure the PnP modern search web part to filter the documents from all the sites based on the above refinable.

but i have the following 2 main questions, if anyone can help me in making decision on them:-

Question-1) is the approach of having 2 site columns with the same internal name inside the 2 sites, but linked to 2 different term sets, with the intention to be able to filter documents from the 2 sites using the same Refinable, a valid approach?

Question-2) now for the Document Type inside the Term Store, we can create 2 separate term sets; "HR Document Type" term set & "Commercial Document Type" term set as shown above , as follow:-

OR we can have one parent term set named "Document Type", and under it to create 2 sub terms (HR & Commercial), as follow, and link the site columns to the sub-terms instead of a term set:-

so which approach we should use ? 2 term sets? or one term set with 2 sub terms? and why?

Can anyone advice on the above 2 questions?

Thanks and i really appreciate any help in advance