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The problem is the use of a nonisolated function to initiate an asynchronous update of an actor-isolated property. (I'm surprised the compiler even permits that.) Not only is it misleading, but actors also feature reentrancy, and you introduce all sorts of unintended races.

However, you can add an actor-isolated function to Limiter:

func submit(task: @escaping () async -> Void) {
    switch policy {
    case .throttle: throttle(task: task)
    case .debounce: debounce(task: task)
    }
}

Note, I am not using callAsFunction as an actor-isolated function as it looks like (in Xcode 13.2.1, for me, at least) that this causes a segmentation fault in the compiler.

Anyway, you can then modify your tests to use the submit actor-isolated function, e.g.:

// test throttling as user enters “hello, world” into a text field

func testThrottler() async throws {
    // Given
    let promise = expectation(description: "Ensure first task fired")
    let throttler = Limiter(policy: .throttle, duration: 1)

    var fulfillmentCount = 0
    promise.expectedFulfillmentCount = 2
    var value = ""

    func accumulateAndSendToServer(_ input: String) async {
        value += input

        await throttler.submit { [value] in
            // Then
            switch fulfillmentCount {
            case 0:  XCTAssertEqual(value, "h")
            case 1:  XCTAssertEqual(value, "hello,")
            default: XCTFail()
            }

            promise.fulfill()
            fulfillmentCount += 1
        }
    }

    // When
    await accumulateAndSendToServer("h")
    await accumulateAndSendToServer("e")
    await accumulateAndSendToServer("l")
    await accumulateAndSendToServer("l")
    await accumulateAndSendToServer("o")

    try await Task.sleep(seconds: 2)

    await accumulateAndSendToServer(",")
    await accumulateAndSendToServer(" ")
    await accumulateAndSendToServer("w")
    await accumulateAndSendToServer("o")
    await accumulateAndSendToServer("r")
    await accumulateAndSendToServer("l")
    await accumulateAndSendToServer("d")

    wait(for: [promise], timeout: 10)
}

As an aside:

1. In debounce, the test for isCancelled is redundant. The Task.sleep will throw an error if the task was canceled.

2. As a matter of convention, Apple uses operation for the name of the closure parameters, presumably to avoid confusion with Task instances.

3. I would change the Task to be a Task<Void, Error>?. Then you can simplify debounce to:

   func debounce(operation: @escaping () async -> Void) {
       task?.cancel()
   
       task = Task {
           defer { task = nil }
           try await Task.sleep(seconds: duration)
           await operation()
       }
   }
   

4. When throttling network requests for user input, you generally want to throttle the network requests, but not the accumulation of the user input. So I have pulled the value += input out of the throttler/debouncer. I also use a capture list of [value] to make sure that we avoid race conditions between the accumulation of user input and the network requests.

FWIW, this is my rendition of Limiter:

actor Limiter {
    private let policy: Policy
    private let duration: TimeInterval
    private var task: Task<Void, Error>?

    init(policy: Policy, duration: TimeInterval) {
        self.policy = policy
        self.duration = duration
    }

    func submit(operation: @escaping () async -> Void) {
        switch policy {
        case .throttle: throttle(operation: operation)
        case .debounce: debounce(operation: operation)
        }
    }
}

// MARK: - Limiter.Policy

extension Limiter {
    enum Policy {
        case throttle
        case debounce
    }
}

// MARK: - Private utility methods

private extension Limiter {
    func throttle(operation: @escaping () async -> Void) {
        guard task == nil else { return }

        task = Task {
            defer { task = nil }
            try await Task.sleep(seconds: duration)
        }

        Task {
            await operation()
        }
    }

    func debounce(operation: @escaping () async -> Void) {
        task?.cancel()

        task = Task {
            defer { task = nil }
            try await Task.sleep(seconds: duration)
            await operation()
        }
    }
}

Which uses these extensions

// MARK: - Task.sleep(seconds:)

extension Task where Success == Never, Failure == Never {

    /// Suspends the current task for at least the given duration
    /// in seconds.
    ///
    /// If the task is canceled before the time ends,
    /// this function throws `CancellationError`.
    ///
    /// This function doesn't block the underlying thread.
    public static func sleep(seconds duration: TimeInterval) async throws {
        try await Task.sleep(nanoseconds: UInt64(duration * .nanosecondsPerSecond))
    }
}

// MARK: - TimeInterval

extension TimeInterval {
    static let nanosecondsPerSecond = TimeInterval(NSEC_PER_SEC)
}

And the following tests:

final class LimiterTests: XCTestCase {

    // test throttling as user enters “hello, world” into a text field

    func testThrottler() async throws {
        // Given
        let promise = expectation(description: "Ensure first task fired")
        let throttler = Limiter(policy: .throttle, duration: 1)

        var fulfillmentCount = 0
        promise.expectedFulfillmentCount = 2
        var value = ""

        func accumulateAndSendToServer(_ input: String) async {
            value += input

            await throttler.submit { [value] in
                // Then
                switch fulfillmentCount {
                case 0:  XCTAssertEqual(value, "h")
                case 1:  XCTAssertEqual(value, "hello,")
                default: XCTFail()
                }

                promise.fulfill()
                fulfillmentCount += 1
            }
        }

        // When
        await accumulateAndSendToServer("h")
        await accumulateAndSendToServer("e")
        await accumulateAndSendToServer("l")
        await accumulateAndSendToServer("l")
        await accumulateAndSendToServer("o")

        try await Task.sleep(seconds: 2)

        await accumulateAndSendToServer(",")
        await accumulateAndSendToServer(" ")
        await accumulateAndSendToServer("w")
        await accumulateAndSendToServer("o")
        await accumulateAndSendToServer("r")
        await accumulateAndSendToServer("l")
        await accumulateAndSendToServer("d")

        wait(for: [promise], timeout: 10)
    }

    // test debouncing as user enters “hello, world” into a text field

    func testDebouncer() async throws {
        // Given
        let promise = expectation(description: "Ensure last task fired")
        let debouncer = Limiter(policy: .debounce, duration: 1)
        var value = ""

        var fulfillmentCount = 0
        promise.expectedFulfillmentCount = 2

        func accumulateAndSendToServer(_ input: String) async {
            value += input

            await debouncer.submit { [value] in
                // Then
                switch fulfillmentCount {
                case 0:  XCTAssertEqual(value, "hello")
                case 1:  XCTAssertEqual(value, "hello, world")
                default: XCTFail()
                }

                promise.fulfill()
                fulfillmentCount += 1
            }
        }

        // When
        await accumulateAndSendToServer("h")
        await accumulateAndSendToServer("e")
        await accumulateAndSendToServer("l")
        await accumulateAndSendToServer("l")
        await accumulateAndSendToServer("o")

        try await Task.sleep(seconds: 2)

        await accumulateAndSendToServer(",")
        await accumulateAndSendToServer(" ")
        await accumulateAndSendToServer("w")
        await accumulateAndSendToServer("o")
        await accumulateAndSendToServer("r")
        await accumulateAndSendToServer("l")
        await accumulateAndSendToServer("d")

        wait(for: [promise], timeout: 10)
    }

    func testThrottler2() async throws {
        // Given
        let promise = expectation(description: "Ensure throttle before duration")
        let throttler = Limiter(policy: .throttle, duration: 1)

        var end = Date.now + 1
        promise.expectedFulfillmentCount = 2

        func test() {
            // Then
            XCTAssertLessThanOrEqual(.now, end)
            promise.fulfill()
        }

        // When
        await throttler.submit(operation: test)
        await throttler.submit(operation: test)
        await throttler.submit(operation: test)
        await throttler.submit(operation: test)
        await throttler.submit(operation: test)

        try await Task.sleep(seconds: 2)
        end = .now + 1

        await throttler.submit(operation: test)
        await throttler.submit(operation: test)
        await throttler.submit(operation: test)

        try await Task.sleep(seconds: 2)

        wait(for: [promise], timeout: 10)
    }

    func testDebouncer2() async throws {
        // Given
        let promise = expectation(description: "Ensure debounce after duration")
        let debouncer = Limiter(policy: .debounce, duration: 1)

        var end = Date.now + 1
        promise.expectedFulfillmentCount = 2

        func test() {
            // Then
            XCTAssertGreaterThanOrEqual(.now, end)
            promise.fulfill()
        }

        // When
        await debouncer.submit(operation: test)
        await debouncer.submit(operation: test)
        await debouncer.submit(operation: test)
        await debouncer.submit(operation: test)
        await debouncer.submit(operation: test)

        try await Task.sleep(seconds: 2)
        end = .now + 1

        await debouncer.submit(operation: test)
        await debouncer.submit(operation: test)
        await debouncer.submit(operation: test)

        try await Task.sleep(seconds: 2)

        wait(for: [promise], timeout: 10)
    }
}