Usually at big shoots, what I have seen is HAA is the person who scored the best cumulative score for the "big 3 events," that is, Event Singles championship (usually 200 targets) Event Doubles championship, and the Event Handicap championship.
Its usually delineated by your singles classification, that is, there will be an Event HAA champ, and Event HAA runner-up, Class AA HAA champ, Class A HAA champ, etc.
A perfect HAA score is usually 400.
HOA is often a the cumulative score of a larger number of events, usually 8 or 10 events comprising of preliminary events and championship events. That's why you'll see HOA totals in the thousands.
In order to score well in HOA, you have to enter all of the preliminary events.
My experience in the far west is that HOA (High Over All) rates shooters combining all their scores in singles, handicap, and doubles that are part of the official shoot. You aren't even entered if you skip a or some events and it never includes "warmup" events before the registered shoot. HAA, (High All Around) as said often above, is your scores for the featured, individual events, singles, handicap, and doubles, determining champions. We often see both offered in programs in fairly ordinary weekend shoots; 500 or 600 total birds.
As was posted before some confusion was added - HAA is the trophy on the combined score of the shoot's Championship events. It does not have to be 400 targets, and the ATA does not define it that way for all shoots. At some larger shoots with ATA trophies, the championships are requred to be 200,100 and 100, that's where the 400 comes from. Many shoots only have 300 total in the championships, so the HAA would be on those 300. Sometimes management has to shorten an event due to turnout, weather etc., in which case the HAA is still based on the championship event targets set by management.
HOA is the total of the championship events plus whatever other events management wishes to include. You have to read the program to know for sure.