The plant is fine and looks good. The small pot is normal as the plant was in a nursery receiving daily water or misting. The broke leaf is OK. If it turns brown or IF you Must just cut it off at the pstem (stalk).
The bigger pot will hold more water in the soil and reduce the need for daily watering. Depending on environmental factors, potted bananas should be OK being watered about every 7 to10 days. Top 2" of the pot soil needs to dry out between watering. Excess Water MUST drain from the pot. Standing or stagnate water in the bottom of the pot promote root rot and will kill the plant. The soil only needs to be moist.
Banana plants are a tropical plants use to high humidity and have large leaves for collection of morning tropical rains & dew. The home environment has a low humidity which is not good for bananas. Misting is need to prevent dehydration of the plant. Mist 2 or 3 Xs per day. ... Heck, I do this with my in ground banana outside.
Potting soil. Generally a mix of 50% commercial potting soil with 50% of perlite is good. I like to see water when poured on top start draining from the bottom in about 15 - 20 sec. for 10" or 12" pot. Contrary to other advice you may receive, Do not add extra peat or compost to the mix. These will hold water like sponge and drown the roots. Banana roots need air! And do not add bricks, rocks, pot/glass shards and the like to the bottom of the pot to save on potting soil.
Fertilizer: .... Use a balanced fertilizer for garden vegetables. These have the minor nutrients needed by the plant. A tablespoon or 2 for small pots; a handful for large pots. Apply at the top of the soil around the outer edge of the pot. Banana plant need potassium, so add about 3 or 4 tablespoon once a month Sulfate of Potash (SOP; potassium sulfate) or murate of Potash (MOP; potassium chloride).
On my potted Banana plants, I all so give a foliar spray of Miracle Grow about every 3 days. ... More info can be found on the forum wiki
Info:Fertilizer - Bananas Wiki ... there is too much to explain here.