My bike EDC:

1x Zipp tube
Lezyne trigger speed drive c02 inflator
1x 16g c02 cartridge
2x black pedro’s micro levers
Snap-On 15mm ‘midget’ combination wrench
Tekton ball-end 4,5 & 6mm hex wrenches
Meqix patch kit
1x pre-sized stainless brake cable
1x Connex 3/32 masterlink
1x presta to schrader adapter
$20 bill

as a commuter myself, I understand the need to be properly prepaired, but also not wanting to drag a bunch of stuff along. This is what I consider essential for day to day commuting. If your commute is quite long, or you are prone to flats, I would strongly recommend some type of frame mounted pump. I’m not recommending you buy all these specific things, but a basic maintenance and flat kit is pretty essential. Personally, I despise multi tools because they are just a PITA to use, especially on the side of the road in the cold or wet. Also, you’re likely to never use 90% of the tools it has. So concentrate on a few tools you know you’ll need for the most basic repairs for your bike specifically and purchase those. I promise it’s much better having a few properly selected purpose fit, quality tools, than an over priced, near useless multi tool. I choose to carry a few actual hex wrenches, because it’s all i need. Beyond that, I think you’ll find most folks basic kits are pretty similar.

Im not sure I can give any worthwhile recommendations on how to carry your tools, because everybody is different and their riding is different. I personally, carry my kit in a small leather brooks frame bag. I think any quality saddle bag or frame bag will do. If you ride with a backpack or shoulder bag, you could just throw your kit in there. The best advice I can give though, I to sit down and think very carefully about what exactly you’ll need for your bike, but keep it simple. Just want you need to change a tube/fix a flat and do a few small repairs/adjustments on your bike if necessary.