From my experience Snug tight + one-third means fully tensioned.
Below is a link to the NYSDOT Steel Construction Manual's section on bolting. I copied the definition of snug tight but take a look because there are some tables on required pretension and proper inspection procedures.
"Bolts shall be tightened to the minimum required tension described in Table 1001.5a by the turn-of- thenut
method. Enough bolts shall be installed and brought to a snug tight condition to ensure that the parts
of the joint are brought into full contact with each other. Snug tight is defined as the tightness attained
by a few impacts of an impact wrench or the full effort of a person using an ordinary spud wrench. This
represents approximately 150 foot pounds [200 N m] for bolts ? inch (M22) in diameter and larger.
Following this initial operation, bolts shall be placed in any remaining holes in the connection and
brought to snug tightness. All bolts in the joint shall then be tightened additionally by the applicable
amount of nut or head rotation specified in Table 1001.5b. During this operation, there shall be no
rotation of the part not turned by the wrench."