Use an outline or old updates as a template
Basically, you're taking minutes. What happened, who did it, and the rough order it happened in (with some grouping by category).
Taking minutes from scratch is pretty difficult, especially if you're active in the session. So start with a framework.
The last update is a good jumping off point. If the characters are mostly the same and they're still on the same quest, a lot of what happens is going to be similar, even if the events are very different.
Here's your sample excerpt from your updates post:
A villain thwarted, a new companion found and a scroll retrieved. A
great deal had happened to the Friday Knights since they left
Sandpoint. What would they be facing upon their return? Sandpoint
would certainly be different than when last they saw it.
What to do first? They had debts to collect and an old wizard to speak
to. Then where to? All their immediate plans were at an end. Only time
would tell what lay ahead...
If you had that next to you as you played, you could simply add notes to the appropriate places. Maybe cross out "debts to collect." Because you didn't do any of that in this session. Then circle "old wizard" and point an arrow towards a few notes in the margin about the wizard and how that visit went. Make a note as well about what Sandpoint was like, or how close they are now, if they're not there yet.
After the gaming session ends, go over the papers and use the old update as a prompt for anything you forgot to write down.
It seems simple but I'm someone who simply can't take minutes of a meeting or I can't focus. But when I ran the meetings and used a copy of the old minutes (since we were planning a large event and we always had something to say about each section), I found it very easy to jot down a few details and had no problem being the minute taker.
Not having to write the topic was a lifesaver. Just a few extra words, but they make all the difference. It also helps you read the notes later, since the topics are typewritten.
I also used the agenda printout to take notes of anything that wasn't in the old minutes. Again, the prompts were very helpful. You may not have a meeting agenda but you certainly will have a list of things you expect might happen. They're right there in your update sample. Add in any extra things that might occur that session and print it up with the rest.
One of the most important parts is to type up your notes the same day or the day after. Don't let them sit too long or you'll forget things. You can make them creative or proofread them another day. But get the content down fast.
Do I really need to use paper? you might be asking. And yes, you probably do. It helps tremendously. If you don't already write your notes down on paper, give it a try. Being able to circle and cross out and use arrows minimizes how much writing you have to do and keeps things in order.