I'm not sure I would characterize it as 'go with the more stringent'.
by contract you would be required to comply with both. If there is a conflict the Code wins. Unless of course the client has no wish to claim it was Code compliant.

Just like when I order a piece of material, I as  the purchaser need to let the supplier know exactly what I want. Sometimes this may be by reference. Normally the purchasing document used to specify the inspections to be provided would give a clear direction to the order of precedence for acceptance criteria.

The acceptance criteria for welder qualification testing is very generic for some codes. If you are testing welders for another organization, be sure you are aware of all of their acceptance criteria. Not meeting it or exceeding can be an issue when you are manning up a job.

Do what the instructions provided for the project spell out. No more, no less.

An owner can (at its option) specify anything they wish except where some variation of law sets forth specific requirements.
Playing the part of owner, I could in theory 'specify' zero tolerance for any measurable flaw or conversely, specify all inspections be performed by uncertified blind gypsy circus monkeys, or for that matter, specify zero inspections be performed.

No code I am aware of assumes liability for products built utilizing any part of the specified code. You can't sue ASME, AWS, or other such for a defective product.

The NRC can't  take action against ASME section III committee, but they can write regulatory guidelines that the owner had to abide by if they wish to operate a nuclear power plant. That's an example of a law enforcing a code. The NRC can specify that parts of the code be enhanced, ignored etc as well.

Regarding the liability part, it's usually a bad idea to write a less stringent specification if the owner wishes to be insured and have some level of protection in a lawsuit should something go wrong. If the specified uncertified blind gypsy circus monkeys miss a crack that causes a catastrophic failure injuring or killing someone at worse, or damaging a product at best, then it is the owner on the hook.

As said, they can specify anything they want within the bounds of the law, but there is also no law requiring you to accept a given contract. It is incumbent upon the owner, not the inspector to address conflicts of this nature in writing. You can always walk if you disagree enough.