Kotlin:

import kotlin.math.abs
import kotlin.math.sign

data class Report(val levels: List<Int>) {
    fun isSafe(withProblemDampener: Boolean): Boolean {
        var orderSign = 0.0f  // - 1 is descending; +1 is ascending

        levels.zipWithNext().forEachIndexed { index, level ->
            val difference = (level.second - level.first).toFloat()
            if (orderSign == 0.0f) orderSign = sign(difference)
            if (sign(difference) != orderSign || abs(difference) !in 1.0..3.0) {
                // With problem dampener: Drop either element in the pair or the first element from the original list and check if the result is now safe.
                return if (withProblemDampener) {
                    Report(levels.drop(1)).isSafe(false) || Report(levels.withoutElementAt(index)).isSafe(false) || Report(levels.withoutElementAt(index + 1)).isSafe(false)
                }  else false
            }
        }

        return true
    }
}

fun main() {
    fun part1(input: List<String>): Int = input.map { Report(it.split(" ").map { it.toInt() }).isSafe(false) }.count { it }

    fun part2(input: List<String>): Int = input.map { Report(it.split(" ").map { it.toInt() }).isSafe(true) }.count { it }

    // Or read a large test input from the `src/Day01_test.txt` file:
    val testInput = readInput("Day02_test")
    check(part1(testInput) == 2)
    check(part2(testInput) == 4)

    // Read the input from the `src/Day01.txt` file.
    val input = readInput("Day02")
    part1(input).println()
    part2(input).println()
}

The Report#isSafe method essentially solves both parts.

I've had a bit of a trip up in part 2:

I initially only checked, if the report was safe, if either elements in the pair were to be removed. But in the edge case, that the first pair has different monotonic behaviour than the rest, the issue would only be detected by the second pair with indices (2, 3), whilst removing the first element in the list would yield a safe report.