Well, first of all, a police certificate is specifically for immigration purposes, so it is what the embassy or high commission will want to receive from you. Many of them specify that they want a police certificate and won't process the application without one.
I'm not sure there is an agreed definition of an active criminal record. After all, anything on your police record, which can also be a criminal record, is there until you are 100, so it can be said to be active. It is just a list of any unspent convictions, and any conditional cautions that have not become spent yet, and how people choose to interpret that is up to them. It can also be used for purposes other than employment, especially insurance if you make a claim. You could, in theory, be asked to produce one for a social housing application. By "qualifying organisation", I'm assuming you are referring to umbrella bodies. They are intermediaries in the DBS check process for organisations that don't process enough themselves over a year. They are registered with the DBS and followed agreed processes to pass on requests for DBS checks. The information they need is the same as the information you need to do a basic DBS check on yourself, but they won't process a check without your consent. That would be a data protection breach. They achieve their status by meeting the DBS's requirements and applying to become a qualifying organisation.
An SAR is a complete record of what is held about you on the Police National Computer (PNC). It lists your name, date of birth, aliases, addresses, appearance and anything relating to being arrested, including whether DNA and fingerprints were taken and whether you were released on bail or under investigation. It then lists what the outcomes (disposals) were, and there is a separate section at the end that lists each disposal in turn. An SAR is what you can do to get your own information. Access is strictly controlled, but there are
organisations who officially have access. You can probably make a separate SAR to ACRO to ask for details of anyone accessing your PNC record. You can also make an SAR to your local police force to ask for any information they hold about you in their files. That is stuff that isn't put on the PNC, such as allegations and intel, and also non-recordable offences. As with any information, it is yours and can't be accessed without your consent, except by people and organisations who have legally been granted access. That does not include foreign embassies and high commissions.
The court transcript can give a bit more context to the case, and will certainly include the judge's remarks, which might be helpful if those remarks are favourable. It is a transcript of what was said, and nothing else. It is also only available for a few years, after which it is deleted. If someone is wanted by the police, they would have trouble leaving the country, and if it is suspected that they want to leave the country, an Interpol alert may well be issued. That wouldn't be communicated to any embassies or high commissions, but the border control of the relevant countries would be aware of it.
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If you are to punish a man retributively you must injure him. If you are to reform him you must improve him. And men are not improved by injuries. (George Bernard Shaw)