Company Evidence - trademark name vs company name

The ACS guidelines states that the payment evidence must cites company name and applicant name.
The company’s name, mentioned in the reference letter, is different from the one appearing in the payment and fiscal documents.

This is because its trademark differs from its registered name.

Is this something common and accepted by ACS or do I need to attach some documentation or evidence that proves that both names belong to the same business?

If this is the case, what documentation should I provide?

In addition the company I’ve worked for has changed its registered name during my employment, so part of my payment evidence makes reference to the old name and some other to the current one.

Also in this case, is this common and if not what documentation should I provide?

If you can get it in writing that all name belong to same company, then you should be fine.