PSE Advent Calendar 2023 (Day 6): It's a beautiful day for a ballgame

Question

<sub>This puzzle is part of the Puzzling Stack Exchange Advent Calendar 2023. The accepted answer to this question will be awarded a bounty worth 50 reputation.

< Previous Door Next Door ></sub>

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It's the annual charity softball game at the North Pole. With this year's game falling on December 6 - Finnish Independence Day - Santa's sleigh-pullers are facing off against a team of reindeer from northern Finland. December has always struck you as a particularly cold time of year for softball, but nobody has seemed to complain. Certainly, nobody in attendance is complaining today, because they're in the midst of witnessing something historic.

Pitching for the North Pole squad today is Olive, the other reindeer, who is no longer hurling insults at her batterymate, Rudolph, and is instead hurling strikes at him. With 2 outs in the top of the 6th and final inning, she has retired every batter she's faced, and now sits just one out away from a perfect game, an astonishing accomplishment. She's worked through some scares, including a tense third inning in which she gave up three deep fly balls to the outfield, but each of her outfielders was in the perfect position that inning - and on the whole, her defense has been in the right position in just about every other inning this game. For most of the day, though, Olive has been in complete control.

As the final batter comes to the plate, everyone in the stands is on their feet, cheering on the home pitcher. The man next to you is dressed in all red - clearly a fan of the home team - and has been diligently filling in a scorecard all day. As Olive throws a pitch - strike two - he seems to have figured something out. He nudges you, points towards the pitcher's mound, and says, "Here comes the hook!" Before you can react, Olive delivers the next pitch, the batter swings and misses, and the crowd erupts. You turn to the man next to you, but incredibly, he's vanished, leaving nothing but his scorecard behind!

You pick it up. You may not be able to figure out what happened to the mysterious man, but maybe you can at least figure out how he seemingly knew what was going to happen before it happened. After several minutes of puzzling over the sheet of paper, you've realized that Olive and crew were in even more control of the game than you thought! Not only did she throw a perfect game, but she finished it off with the perfect pitch for the moment - and not only that, she and the team had telegraphed the whole thing!

So now, I ask you: what was the final pitch Olive threw, and why was it so appropriate for the occasion?

_{Click to enlarge
If you find the scorecard difficult to read, you can access a text version of the scorecard here}

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Hint 1:

For exactly one-third of the visitors' turns at bat, you'll need to interpret the result of the at-bat more literally.

Hint 2:

Any play made by only one fielder (that is, F[#], P[#], and U[#]) can be considered identical for the purposes of this puzzle.

Hint 3:

Each at bat represents a single letter. You should end up with an 18-letter phrase from each half of the scorecard; one which is 3 words long and one which is 5 words long. (Hint 3.5: one at bat is missing from the scorecard, but is mentioned in the preamble.)

Hint 4:

For the two remaining unsolved encodings: one depends on who, the other depends on when.

Answer 1

Final answer after significant help from @juicifer and @one

The final pitch was

a TWELVE SIX CURVEBALL which was appropriate for the 6th of December

Because

each at bat taken in order resolves to a single letter. The method of resolution varies according to outcome and team at bat.

To start with,

strikeouts for the visitors identify the letter in the batter’s name which does not form part of an anagram of a word which is a synonym for either SWING or LOOK accordingly. For the home team strikeouts are the letter K as is usual in batting sports.

Thus

STUYVER = SURVEY* + T; AUSTER = STARE* + U; SWEASER = SEESAW* + R; KOLBO = LOOK* + B; WIGANS = SWING* + A; LAWSY = SWAY* + L and out thirty-six letters now read T?????????UR??BA?L ???K??????????????

Next

the assisted putouts on both sides match up with the visual tag, as treating the player positions as a dot-to-dot puzzle gives capital letter-like shapes e.g. 7-5-6-2-3 ~ "W", "9-7-6-5-3" ~ "E"

Making the

letters TWE?VE????URV?BALL and the second set L??K????????ER?I??

With help in the comments from @one we note that

if we interpret a fielder's catch in inning $x$ as the $x$th letter of the fielders name we get TWELVESIXCURVEBALL and L??KATTHE?THERSIDE.

Finally, we take

a hit for the home team as O and the full thirty-six letters are TWELVE SIX CURVEBALL and LOOK AT THE OTHER SIDE.

As (I hope) that they say in Finland

hyvää joulua kaikille ja hyvää yötä kaikille.