Mulch and compost are two different things, although the exact same material can be used for either purpose (and people get them confused with each other regularly).

Mulch is solid matter, usually but not necessarily organic, spread around the bases of plants to discourage/smother the growth of weeds (i.e., other plants), while also helping to hold moisture in the ground.

Compost is decomposed (or decomposing) organic matter used as fertilizer. Common practice is to maintain a compost heap, and to spread the resultant decomposed matter after it has broken down; however, it is also possible to work not-yet-rotted organic material into the soil, and that can also be called compost.

Grass clippings and other yard-and-garden waste like fallen leaves, etc., are useful for either or both purposes. Organic matter put down as mulch in the spring can be turned under in the fall as compost.

The so-called "mulching" lawnmower is actually designed to transform clippings into instant compost, deposited in place right on the lawn from whence they came. The extra chopping-up produces tiny pieces of plant matter, small enough to fall between the freshly cut blades of live grass directly onto the soil, where it will quickly and invisibly rot and fertilize the lawn. (The smaller the pieces, the faster the decomposition.)

If you want to use cut grass as mulch for your garden, the mower's "mulching" feature would be best turned off. The clippings you collect in your bag, or rake up, really should be as large (coarse) as possible, so the matter does not decompose too quickly, but rather retains its structural integrity as long as possible, all the way through the current growing season.

(Some sources recommend against using freshly-cut, still-green grass clippings as mulch because the early stages of decomposition generate so much heat that the plants around which it is spread can be damaged.)

My mower has three possible settings: rear-bag, side-discharge, and mulching. I usually use the rear-mounted bag to collect clippings that I dump into my compost pile. There's a spring-loaded panel on the back on the mower that lifts up to create an "output port" when the bag is attached, and snaps shut when the bag is not in use. There's another spring-loaded opening on the right side; it is held open by the "chute" attachment when you want the old standard "side-discharge " arrangement. When no accessory is attached and both the rear and the side openings are closed, it's a mulching mower ~ the clippings get chopped repeatedly while trapped under the mower.

I never use the side-discharge method, and indeed don't even know where I put that attachment that I've never used and never will.

My main reason for occasionally going into "mulching" mode is to avoid having to change and empty the bag too often, when cutting really high grass and weeds. (When I first bought this mower, we were the first occupants of our block after Katrina, and I cut all the front lawns after weeds had grown very high. I still occasionally mow a long-neglected yard, ditching the bag when I do so.)