no need to create ContentType and LinkedResource objects. Add the embedded image as an attachment. Also, the EmbedImage method expects the name of the image, not the full path.
:
var myMessage = new SendGridMessage();
contact_list = new List<MailAddress>();
contact_list.Add(new MailAddress("email@gmail.com"));
myMessage.To = contact_list.ToArray();
myMessage.From = new MailAddress("clientservice@stpis.com");
myMessage.Subject = "Subject";
string html = "<div><img src=cid:Logo /></div>";
myMessage.Html = html;
**myMessage.AddAttachment(@"C:\logo.png");
myMessage.EmbedImage("logo.png", "Logo");**
SendMessage(myMessage);