Reiki is not a form of massage. It can be added to a massage, but it is energy work. Sports massage and deep tissue are the same thing. And trigger point therapy is part of deep tissue. Swedish is just your basic, "make me feel relaxed" massage. Lomi lomi is more of a healing technique than anything else and it's kind of intense; I wouldn't recommend it for a first-time massage. You lay directly on the vinyl of the table and are totally naked except for a towel, whereas in any other massage there are blankets and sheets and cushioning and heating pads. It also works with the energetic body as well as the physical body. For hot stone massage, as the name implies, the therapist massages with basalt stones that have been heated. It's really good for conditions like arthritis that respond well to penetrating heat, but it's another one that I would not recommend for a first-time experience.
For a muscular man, such as one who is in the military, he's probably going to benefit the most from deep tissue, partly because anyone who is physically active on that kind of level needs some serious care on their muscles (which is why it's also called sports massage), and partly because men tend to be more comfortable with the idea of a massage when they know it's doing something they can feel.
Also on the subject of men being comfortable with massage, most of the men I have seen come in for a first massage seem to be more comfortable if they are not alone. I suggest you set it up as a couple's massage and enjoy it with him. You deserve it, too; holding down the fort while he's deployed is pretty stressful.