Dart

Finally got round to solving part 2. Very easy once I realised it's just a matter of counting line crossings.

Edit: having now read the other comments here, I'm reminded that the line-crossing logic is actually an application of Jordan's Curve Theorem which looks like a mathematical joke when you first see it, but turns out to be really useful here!

var up = Point(0, -1),
    down = Point(0, 1),
    left = Point(-1, 0),
    right = Point(1, 0);
var pipes = >>{
  '|': [up, down],
  '-': [left, right],
  'L': [up, right],
  'J': [up, left],
  '7': [left, down],
  'F': [right, down],
};
late List> grid; // Make grid global for part 2
Set> buildPath(List lines) {
  grid = lines.map((e) => e.split('')).toList();
  var points = {
    for (var row in grid.indices())
      for (var col in grid.first.indices()) Point(col, row): grid[row][col]
  };
  // Find the starting point.
  var pos = points.entries.firstWhere((e) => e.value == 'S').key;
  var path = {pos};
  // Replace 'S' with assumed pipe.
  var dirs = [up, down, left, right].where((el) =>
      points.keys.contains(pos + el) &&
      pipes.containsKey(points[pos + el]) &&
      pipes[points[pos + el]]!.contains(Point(-el.x, -el.y)));
  grid[pos.y][pos.x] = pipes.entries
      .firstWhere((e) =>
          (e.value.first == dirs.first) && (e.value.last == dirs.last) ||
          (e.value.first == dirs.last) && (e.value.last == dirs.first))
      .key;

  // Follow the path.
  while (true) {
    var nd = dirs.firstWhereOrNull((e) =>
        points.containsKey(pos + e) &&
        !path.contains(pos + e) &&
        (points[pos + e] == 'S' || pipes.containsKey(points[pos + e])));
    if (nd == null) break;
    pos += nd;
    path.add(pos);
    dirs = pipes[points[pos]]!;
  }
  return path;
}

part1(List lines) => buildPath(lines).length ~/ 2;
part2(List lines) {
  var path = buildPath(lines);
  var count = 0;
  for (var r in grid.indices()) {
    var outside = true;
    // We're only interested in how many times we have crossed the path
    // to get to any given point, so mark anything that's not on the path
    // as '*' for counting, and collapse all uninteresting path segments.
    var row = grid[r]
        .indexed()
        .map((e) => path.contains(Point(e.index, r)) ? e.value : '*')
        .join('')
        .replaceAll('-', '')
        .replaceAll('FJ', '|') // zigzag
        .replaceAll('L7', '|') // other zigzag
        .replaceAll('LJ', '') // U-bend
        .replaceAll('F7', ''); // n-bend
    for (var c in row.split('')) {
      if (c == '|') {
        outside = !outside;
      } else {
        if (!outside && c == '*') count += 1;
      }
    }
  }
  return count;
}