You can't "split cables". All wiring must be grouped (minimum pair hot/neutral or switched hot/neutral, but sometimes three = hot/travelers or switched hot/travelers etc.). This isn't just to keep track of things, though it does help with that. It is because of the way electricity (alternating current) works. The end result is either cables (2 or 3 or more wires together inside one outer sheath) or conduit (metal or plastic tubes with individual wires inside them. You have cables, and switching to conduit generally doesn't make sense, so stick with cables. But except for getting small pieces to use as pigtails inside junction boxes, you can't use wires from cables outside of a completely assembled cable.
Numbering of wires in cables normally excludes the ground wires. Ground wires are either green or bare. Simply put, ground goes everywhere, all grounds connect together and all devices connect to ground (through a screw, a captive wire or through a metal yoke on a metal box). So the following explanation ignores grounds, and some references in the question/comments to "3 wires" were, I am pretty sure, really what are referred to as "/2" cables = black + white + ground.
Now on the specific problem.
You have a classic switch loop. Confusing when I first learned about them, but not too hard in the end. A switch loop (historically, now a little different with neutral required in switch boxes) with standard cables uses black for switched hot and white (which is normally neutral) for hot. It didn't use white for neutral because the cable only had two wires and a simple switch doesn't need neutral. Actually, based on your diagram, your original switch cable may have been backwards, but it doesn't matter.
You have a duplex receptacle with the top controlled by the switch and the bottom not switched.
You have an incoming (from the panel or previous devices) hot/neutral cable.
You have one additional cable, going on to other places.
The key here is figuring out which cable is the switch cable. Based on your diagram, it appears to be the bottom cable. There are ways to figure it out for sure, but I am assuming that you are 100% certain which cable is the switch cable. So here is what you need to do:
In the receptacle box:
- All 3 cables should have their blacks together with a wire nut. This is the hot bunch.
- All 3 cables should have their neutrals together with a wire nut. This is the neutral bunch.
- A pigtail (short white wire) from the neutral bunch to a silver screw (long slot, left side if you are looking at the front). This side should still have a metal tab connecting the two screws.
- Two pigtails (short black wire) from the hot bunch to the top and bottom brass (short slot, right side) screws. This is critical because the tab was removed from the hot side during the original installation. Alternatively, you could install a new duplex receptacle and then only need one wire on the hot side.
In the switch box:
Assumption: you are using an ordinary (not: smart, timer, motion detector, dimmer, etc.) switch. If you are using any of those then neutral needs to be connected.
YOU MUST RUN A NEW 2-WIRE CABLE FROM THE SWITCH BOX TO THE LIGHT FIXTURE.
IF YOU ACTUALLY CUT UP THE EXISTING CABLE, YOU MUST RUN A NEW 2-WIRE CABLE BETWEEN THE RECEPTACLE BOX AND THE SWITCH BOX.
- Black from old cable goes to one switch screw.
- White from old cable is connected to the new cable white.
- Black from the new cable goes to the other switch screw.
In the light fixture box:
- Black from the new cable goes to the hot screw or black (or red or blue, but usually black) wire.
- White from the new cable goes to the neutral screw or white wire.
Sorry, you will need a new cable between switch and light. That may result in some drywall work, depending on accessibility. In my house it was done (not by me) in 3 rooms without any drywall work, but it helped that the ceiling is accessible in the (otherwise useless) attic.