I've never seen an established lexicon for recording Settlers games, but I bet you could gin one up pretty quickly that would be able to capture everything of note. Here are the major things I think you'd need to capture and my ideas on some compact notation
Board setup. The tiles types are randomized, but the letters always play out in the same order. I'd just create a list with all the letters and then in table format list the tile time under that letter during your setup. Insert a row with no letter for where the desert is located. For example
From here on out, you can use the letters as the signposts for where the action is happening. For example:
- Intersection pieces (e.g. cities) are recorded by the letters of the three surrounding tiles.
- Roads only require two letters.
Then number the players by starting order and record the staring information (i.e. settlement location). Now just record the bare minimum of action as the game progresses. For example:
Turn 1, Player 1
No need to record what each player drew as that can be determine based on the board setup. Record any trades. Ports can just be player 0, where the ratio will tell you if it was a generic or specific.
P4: 1S, 1O; P3: 2W
Record any builds:
Since the information to be captured is going to be exactly the same each game, I bet you could create a blank spreadsheet or table to help ensure you didn't forget anything important. It could have an area for the setup information and then columns for each player and each turn (that could just continue as long as required by the game).
I didn't really discuss the private knowledge aspect of Settlers: the cards. This represents the most challenging aspect to capture in the game log because cards are private knowledge. (I'm assuming you don't have someone not playing who is dedicated to recording the game.)
In this case I suggest recording in the game log when a given player obtains a card, and then having each player record the cards they pull in order. Once the game is over, you can augment the official log with the notes from each player.