Nobody is holding the whole story
It is tempting to assume that somewhere a single file exists with your case in it. On a multi-stage route it does not. Each office sees the paper in front of it, does one job and passes it on, and none of them holds a view of the road from end to end. The only place the full sequence lives is wherever you choose to write it down.
Where Malaysia stands
Malaysia has not joined the Hague Apostille Convention, so papers intended for Malaysia need embassy or consular legalisation instead. There is no apostille issued in Belfast for use in Malaysia that a desk there will act on. The practical consequence is length: more stages, more hand-offs, more dates worth remembering, and a longer gap between sending something and hearing anything about it.
What to write down, and when
One line per document, filled in as you go. Title exactly as printed, issuing office, date on the face of the paper, date it went onward, and any reference given back. Written at the time this costs seconds; reconstructed a month later it costs an afternoon and is usually wrong. An apostille from United Kingdom to Malaysia would have collapsed all of this into a single entry, which is precisely what a longer road takes away.
The count you will be asked for later
Charges on these roads are counted per document, so the number of papers is the number that drives the bill. The published apostille figure is £45 for each document by standard post, or £40 for each on the next-day service open to registered businesses. Keeping a running count as items go in means the total never surprises you, and it also shows plainly when the list has quietly grown past what was actually asked for.
The two offices worth naming in your notes
Write both down properly. The General Register Office for Northern Ireland at Colby House, Stranmillis Court, Belfast BT9 5RR issues certified copies of births, marriages and deaths registered here, because Northern Ireland keeps its own civil register. The FCDO Legalisation Office at PO Box 6255, Milton Keynes MK10 1XX issues apostilles for the whole United Kingdom by post, and searches for what you need for an apostille for Malaysia lead there even though the destination works to another rule.