LOA = Letter of Assurance = a spot for you is reserved IF you become 3Q with a nomination.
LOE = Letter of Encouragement = you appear to be a good candidate who is LIKELY to earn a spot IF you become 3Q with a nom. Because of that, you are authorized an Overnight Visit. These visits are normally reserved for candidates with an offer, but since offers are going out later, this gives individuals with a LOE the opportunity earlier so they can make a quicker determination if offered.
LoQ = Letter of Qualification = you are Academically, Physically, and Medically qualified (3Q) and need just a nomination in order to compete for an appointment. Your file was reviewed by your RC, a random member of the Admissions Committee, and another RC before being certified by the Director of Admissions. This process can take awhile, especially if there are any special reviews that need to be added such as legal checks for suspensions/tickets, or at risk test scores. Sometimes there are also 2Q letters - academically and physically qualified pending medical and a nomination.
As a general rule, expect an RC to work through his/her complete, potentially qualified candidates in order based on the combination of 1) when they finished, 2) likelihood of gaining admission, and 3) whether or not they have a nomination. If you finished early, have a nomination, and are a strong candidate for admission, your file will move to the front of the line. If you finished early, but are weak acadmically or might have an issue to qualifying your file, expect that your RC will hold off on your file to work higher priority applications. Keep working to improve though as he/she will get to it and that extra effort could move you up in the queue.
Each letter means something slightly different. None is an offer. I put them in precedence of likelihood of earning an offer, but the requirement still stands no matter what - you have to be 3Q with a nomination to gain admission. The only letter that means anything in the bigger picture is your Offer, ie, the BFE, that will come in the mail.