Indeed it does seem fairly clear cut, and those conditions are there for a very good reason. The "personal, domestic or household use" part will be derived from Consumer Credit code provisions. There are protections available to consumers (hardship provisions and the like), which only apply to consumer credit, which mean that banks have to treat consumer and business accounts differently.
There have been comments about "why do the banks care if it is business spend - it is all the same as far as they are concerned". Actually, not the case. We are talking credit accounts here, and different types of lending have to be classified differently, and have different capital adequacy requirements, hence the cost of a "business" credit account to the bank may be far higher in cost of capital than a "consumer" account.
The bank cannot turn a blind eye to the transaction types, as it is then risking not complying with prudential requirements. Small amounts are not so significant, as the actual classification probably only covers "predominantly", and the whole exercise is about risk - but make the majority of your spend business expenses on a consumer account at your own peril.
Amex is a totally different beast, as it is not a bank, so totally different rules apply - with much more ability for Amex to manage their risk as they see fit (there being no depositors being protected by the banking regulations).