Some good basic motorcycle oil info.:
http://www.thumperfaq.com/oil.htmYour dirt bike has an integrated engine/transmission, so you are best off using a Motorcycle Specific oil that is API SG/JASO MA. Motorcycle oils contain high pressure additives for your transmission, that "car/diesel" oil don't (can't) contain. And, some car oils have additives like moly that will make your clutch slip, so best to just stay away from car oils completely.
The oil in question may not say API because it isn't API certified. A lot of small oil blenders don't get certified because it is an expensive process. That doesn't mean that it is bad oil, but on the other hand they might not be API because they can't meet the standards (?????).
Your owners manual no doubt recommends Honda GN4, which is a basic petroleum oil (API SJ/JASO MA). My personal preference is Maxima.
http://www.maximausa.com/ They have an inexpensive petroleum oil, a synth blend, and two ester based synthetics. Their add pack is incredible! I run nothing but Maxima Extra ester synthetic in my 8 dirt bikes/ATVs.