A) Your 1988 B230F is LH 2.2. The pages you were looking at in the Volvo wiring manual that Bill Garland copied are for LH 2.4. In 1989, the turbo was still LH 2.2, so go up a couple of pages for the B230FT wiring which will be similar to your year.

B) The ECU has no control over the starter motor and has no idea whether or not it's even cranking. In the Volvo wiring diagram manual, the Starting Circuit diagram is on different pages. Bill's site only has the ECU and ignition pages. The full 1989 green manual (not 1988) can be found online with a bit of effort. I recall there's one in the technical archives at https://ozvolvo.org/archive/

C) You've pretty much proven it's not the starter or solenoid by being able to apply 12 volts directly to the starter and get it to crank.

D) The initial suspect should be a worn ignition switch. Often times a worn ignition switch becomes intermittent and rocking it back and forth or pushing/pulling the key while turning it will get it to crank until it's completely worn out and gone. If it's not intermittent then you can do exactly as Bill said. If you apply 12 volts to that purple wire spade connector with the ignition switch on in KP-II and the engine now cranks then it's the ignition switch and/or battery connection (hence why Bill said to check the battery connection first).

Another and often easier way to diagnose the starter circuit is using a multimeter at that purple spade connector. With the other test lead on chasssis ground, when the ignition is turned to the KP-III start position, you should momentarily see 12V otherwise the ignition switch is gone and/or battery connection.

E) If it's not the ignition switch then the next most likely candidate is the start inhibit (PNP) switch on the shifter console. As a quick check of that switch, try starting the car in both Park and Neutral to see if it's just one side of the switch that's gone. Also try pushing the lever all the way forward in Park while trying to crank. To properly test the start inhibit switch, with the ignition off and again with your meter still on the spade connector, use the ohmmeter function with the other test lead on the blue-green starter solenoid wire. In both Park and Neutral you should have continuity, but not other gears. The PNP switches are a bit of a pain to get at and remove without breaking the plastic studs, so work the circlips off very carefully. Once safely off, you can often restore them by prying them open and cleaning up the copper contacts, otherwise they need to be replaced. They're supposed to be lightly lubed, such as with dielectric silicone grease or even vaseline. You can temporarily bypass the PNP switch by running a wire from the test connector to the blue-green wire terminal on the starter.
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Dave -still with 940's, prev 740/240/140/120 You'd think I'd have learned by now