Generally hay comes in seed hay, meadow hay and rye grass hay. Seed hay tends to be higher sugar/starch as it has the seeds attached, meadow tends to be lower sugar but can have quite a lot of weeds and rye tends to be the highest sugar. It is possible to get Timothy hay and haylage but it is reasonably hard to come by. It tends to be the lowest in sugar.
However, the conditions at the time the hay/haylage is cut can have a big impact on the amount of sugar. You also get first and second cuts which will affect the amount of sugar in the hay.
We had some rye grass haylage a while back for a few that needed some extra condition. It took just 2 weeks on that to have the horses looking really well, and a couple that were ok, but sharing the same field as the skinny ones, looking too fat! Meadow grass hay and haylage works well for our lot generally and they are in medium work - polo two or three times a week and exercise which consists mostly of canter on the days they aren't playing polo with 2 days off a week.