I have seen and had to work with several systems for handling this situation and, as others have said, three is no single answer.
I have seen equipment that had dozens and dozens of PCBs or modules, each with one or more connectors. The schematic for one PCB or module could cover a fold-out page or even two or three of them. Not only did the draftsman have to handle the IN/OUT connectors but also net labels to carry the signal from page to page. It helped a lot to have a good sense of organization: not just left to right, but also top to bottom in many cases. Such drawings seemed to work best if IN/OUT connector(s) were placed in a standard position on all the PCB or module schematics. Or, it was easy to understand what was happening if a connector that carried both IN and OUT connections was broken up into two halves, one on the extreme left of the first page and the other on the extreme right of the last. And signals that went to somewhere in the middle were carried across with complete lines (except for jumping from one page to the next). Just label them both as J1 and show dotted lines up to an S curve to indicate the break on each half. This would be on the bottom of the left half and the top of the right half. If three segments are needed then the middle one would have the break indication on both top and bottom. It kind of shows the flow.
But that's an extreme example. It may have been just my bad luck to meet with it so many times.
If your schematic fits on one page, then I would say there would be no harm in placing half of the combined signal connector on the left and the other half on the right. Again just label them both as J1 and show dotted lines up to an S curve to indicate the break on each half. That would be my choice. Oh, and you don't need to get pictorial about all this. That may be OK for connectors with a small number of pins, but when there are dozens and dozens of them, just simple blocks with numbers in them are better. I am working on a device with a DA-15 connector and I did choose to draw the pins in their actual pattern. That is about as many as I would do that with. I am hoping it will help with the circuit board layout where I intend to use strip board. It's a one-off.
Your numbers:
2. So you would have the connector surrounded by regular lines for connections on one side and net labels on the other. I don't like that.
3. Net labels for every connection? So almost none of the connections would be made with a full line? Definitely not. I have had to work with drawings like this: they stink! This would make understanding the circuit almost impossible. The very idea of a schematic, unlike a pictorial of the device, is to make the operation of the circuit EASY to understand. Or, at least as easy as possible. Don't take it in the opposite direction.
4. The connector pins all over the drawing? Not just no, but HELL NO! Worse than #3.
That's my take on it and I have seen an awful amount of schematics. Had to work with them and figure out what was going on. Make it easier, not harder.
PS: This is a great question for a beginner to be asking. This is the beginner forum, isn't it. Stuff like this should be taught BEFORE Ohm's law.