What do the terms “Grass-fed” and “Grass-finished” mean?

Below is the official USDA  claim standard.
Grass (Forage) Fed – Grass and forage shall be the feed source consumed for the lifetime of the ruminant animal, with the exception of milk consumed prior to weaning. The diet shall be derived solely from forage consisting of grass (annual and perennial), forbs (e.g., legumes, Brassica), browse, or cereal grain crops in the vegetative (pre-grain) state. Animals cannot be fed grain or grain byproducts and must have continuous access to pasture during the growing season. Hay, haylage, baleage, silage, crop residue without grain, and other roughage sources may also be included as acceptable feed sources. Routine mineral and vitamin supplementation may also be included in the feeding regimen. If incidental supplementation occurs due to inadvertent exposure to non-forage feedstuffs or to ensure the animal’s well being at all times during adverse environmental or physical conditions, the producer must fully document (e.g., receipts, ingredients, and tear tags) the supplementation that occurs including the amount, the frequency, and the supplements provided.
Grass (Forage) Fed Marketing Claim Standard (October 16, 2007, Federal Register Notice (72 FR 58631)) – PD
At Buffalo Hills Ranch “grass-finished” means that animals are allowed to accumulate a layer of back fat and varying amounts of marbling within the muscle. This fat in grassfed animals with its more desirable saturated fatty acid profile provides valuable taste, aroma and tenderness components to the eating experience.