The government orders rate is typically based on government per diem, set by the general services administration. As this rate can be vastly below BAR, hotels have the right to ask for proof of government travel at check in. If it explicitly states it is not for leisure travel, there is a possibility the rate could be pushed up at check in.

The military family rate is a discount off of best available rate. The honors rate you are seeing is probably something like the dream away rate which is currently a larger discount off BAR to encourage travel while travel demand is so severely limited. In regular times, military family rate is usually about the same or better than AAA or AARP.