There are 2 parts to your question and each has a distinct answer....
Part 1: What is a registrable pecuniary interest?
Answer: As defined in the second column of the schedule - https://www.legislation.gov.uk/uksi/2012/1464/contents/made
Part 2: Does it come under code of conduct?
Answer: That very much depends on what is / isn't being done.
In relation to disclosable pecuniary interests you will commit a criminal offence (strict liability offence), if, without reasonable excuse, you (i) fail to provide information that you are required to register; or (ii) provide information that is false or misleading or you are reckless as to whether the information is true and not misleading. If you commit such an offence you can be liable on summary conviction to a fine not exceeding level 5 on the standard scale and be disqualified from holding public office for up to 5 years. *S34 Localism Act 2011.
Now you 'could' argue that failing to disclose a DPI, whether convicted or not, could be interpreted as bringing one's self and office into disrepute and preventing someone who is entitled to access information - both of which 'could' be considered under CoC procedures - the big but being - you have to get the MO to acknowledge / agree that the potential offence is valid.
Recently, I identified a Cllr that had not registered land that he owned within the parish that was subject to planning application, had not registered that his company held the grounds maintenance contract for the parish (£+/-15k pa) and that there were various other anomalies within his DPI register held by the MO.
Since I knew the MO would unlikely take it with the seriousness it deserved, I forwarded it direct to the local constabulary. That certainly caused the MO to pull his socks up - however, the constabulary stated that, since it was not a recordable offence they would not prosecute - the obvious flaw in that rationale being that neither are traffic offences recordable but they still prosecute them. The MO (the LA actually) does have power to prosecute but, via the MO, choose not to.
So what do you actually have to do to get a Cllr prosecuted? This was, in all respects, a slam dunk, open and shut, strict liability (see definition below) offence yet the MO refused to act. Would make the Liss Truss lettuce look strong....
* strict liability offence: Strict liability crimes are crimes which require no proof of mens rea in relation to one or more aspects of the actus reus. Strict liability offences are primarily regulatory offences aimed at businesses in relation to health and safety. Also many driving offences are crimes of strict liability eg.
