Cookie created by API client not seen by server

Question

I am writing tests for an API client in C# (i.e. the client is not a web browser). The API server (which I am also in control of) can accept the access token in either the HTTP body or via a cookie. I want to be able to test submitting the access token via either the body or a cookie for any given endpoint. I could do all the cookie tests first, then reauthenticate and do all the access-token-in-body tests (I have gotten this working with slightly different code to what is below) but I prefer not to have to run the tests in a specific order. So my plan is:

1. make a request to the authentication endpoint and tell it that the access token is to be returned in the message body only (no cookie response)
2. create a cookie on the client side using the access token
3. for the cookie tests, make requests to endpoints using the cookie created in 2
4. for the access-token-in-body tests, make requests to endpoint without sending a cookie

However when I do step 3 and inspect the Request.Cookies object at the server side, no cookie is being received (.Count == 0).

Is my code wrong, or is what I am trying to do here not allowed by the spec?

E.g. maybe the spec only allows cookies to be received if they were created by the server?

The endpoint for step 3 above is a standard System.Web.MVC.Controller method and I use an attribute to validate the access token like so (simplified to only show relevant code):

namespace myServerProject.Controllers
{
    public class myController : Controller
    {
        [AcceptVerbs(HttpVerbs.Options | HttpVerbs.Post)]
        [Route("myEndpoint")]
        [myTokenFilter]
        public ActionResult MyEndpoint()
        {
            // do stuff and return result
        }
    }
}

namespace myServerProject.Filters
{
    public class myTokenFilter : ActionFilterAttribute
    {
        public override void OnActionExecuting(ActionExecutingContext filterContext)
        {
      
            string accessToken = HttpContext.Current.Request.Cookies["accessToken"].Value;

            // do stuff and add accessToken to filterContext
        }
    }
}

But HttpContext.Current.Request.Cookies is coming up empty at the server end.

Client side code:

    public CookieContainer Post(
        string url,
        CookieContainer cookies // System.Net
    )
    {
        // new connection each time
        CookieContainer cookieContainer = new CookieContainer();
        HttpClientHandler handler = new HttpClientHandler() // System.Net.Http
        {
            CookieContainer = cookieContainer
        };
        HttpClient httpClient = new HttpClient(handler); // System.Net.Http
        cookieContainer = cookies;

        response = httpClient.PostAsync(url, null).Result;
        string resultString = response.Content.ReadAsStringAsync().Result;
        return GetAllCookiesFromHeader(response.Headers, "myHostname");
    }

Note: GetAllCookiesFromHeader() is defined here and gets the cookies out of the response, ready to be passed back in as a Post() parameter on the next call.