What you are trying to do is impossible, because services are unaware of host names. You might be able to reach what you want by modifying nginx configuration within the pod.
What you need to do is to add an ingress resource. Ingresses are aware of host names. They do load balancing between services. So, /site1 one would go to one service, and /site2 to the other one.
This is what the ingress should look like:
apiVersion: extensions/v1beta1
kind: Ingress
metadata:
  name: ingress
spec:
  rules:
  - host: site-one.mydomain.com
    http:
      paths:
      - backend:
          serviceName: nginx-service1
          servicePort: 80
  - host: site-two.mydomain.com
    http:
      paths:
      - backend:
          serviceName: nginx-service2
          servicePort: 80
As you can see site1 and site2 would be running in different deployments, with different services targeting them. Ingress, which is L7 Load Balancer would be able to check the host name and forward the request to the right service, which in the end would hit the right pod.
You can also add paths to each host, so site-one.mydomain.com/path1, site-one.mydomain.com/path2 and site-two.mydomain.com/path1 , would be forwarded to different services.
The Ingress yaml file would look like this:
apiVersion: extensions/v1beta1
kind: Ingress
 metadata:
  name: ingress
 spec:
  rules:
  - host: site-one.mydomain.com
    http:
      paths:
      - path: /path1
        backend:
          serviceName: nginx-service1
          servicePort: 80
      - path: /path2
        backend:
          serviceName: nginx-service2
          servicePort: 8080
      - path: /
        backend:
          serviceName: nginx-service3
          servicePort: 80
  - host: site-two.mydomain.com
    http:
      paths:
      - path: /path1
        backend:
          serviceName: nginx-service1
          servicePort: 80
Try this out and let me know if it worked for you!