This applies to the EU only
Own Brand Labelling went away as an option in September 2017. The legal manufacturer now has 100% requirement to retain and manage all documentation (etc), even if they are a virtual manufacturer.
"Manufactured By" and "Manufactured For" were never legally recognised under the MDD. I think these exist to some degree under the FDA QSR.
You say you "manufacture for". That doesn't really mean anything under the EU MDD since there can only be one legal manufacturer, and that legal manufacturer must have its own EU Rep if the LM is outside the EU.
You are either
i) a subcontractor to the legal manufacturer (your customer), or
ii) you are the legal manufacturer, and your customer is formally distributing your product.
There isn't a third option.
Things get a little murkier, since you could potentially have a big "marketer name" and a small "legal manufacturer" name (next to the EN ISO 15223 symbol for legal manufacturer).
But that's probably a little risky, since as I mentioned upthread, the MHRA really doesn't like that, and that's a view that's probably shared by the other Competent Authorities.