Check your ignition switch (the electrical switch mounted on the back of the ignition lock cylinder). The ignition switch supplies +12v to the coil in both run and the crank key positions. It almost sounds like sometimes the coil is not getting +12v in the crank position and as soon as you return the key to the run position 12 v is applied to the coil and the engine starts. This is a fairly common problem associated with a dirty or worn ignition switch (and the flip problem where the car starts in the crank position; but, dies when you return to the run position).
I have a 142 E so I don't have the 1800 wiring diagram. You need to go here to get a copy of the 1800 wiring diagram if you don't have it already.
http://volvo1800pictures.com/sweden/Volvo_1800_dokumentation_main_page_en.php
From the wiring diagram, you will be able to find the contact off of the ignition switch which supplies the ignition coil. Disconnect the starter solenoid connection (you don't need to be cranking the hell out of your starter while doing this test!) and then check to make sure that you are getting 12v on that contact in both the run and crank positions. Since this appears to be an intermittent problem, try wiggling the key and moving it back and forth between run and crank to see if you can initiate failure of the switch. On my 140, new switches are available and not that expensive. Since the 1800 is more popular and typically has more reproduction parts than the 140, I would have thought replacement ignition switches would have been a no brainer. However, if the switch is similar to the 140 switch, you can open up the switch by bending back a couple of metal retaining tabs on the metal cover (watch for parts springing / dropping out). This gives you access to the switch contacts and sometimes you can fix the problem up by cleaning the contacts and applying a little of that Oxgard grease.
If you search on the internet, I seem to recall seeing a post by someone who took their 1800 ignition switch apart and rehabilitated it. I think it was on an British Volvo forum.
I rehabilitated my 142 E after a close to 35 year storage. Bad electrical connections were a huge issue. I went through every electrical connection cleaning and tightening up loose spade connections and lug/stud connections and checking the condition of the wires. Since you have a D jet, be aware that the insulation on the wires where they enter the termination plugs on the 4 fuel injectors, the throttle position switch, the engine coolant sensor and the intake air temperature sensor tends to harden, crumble and fall off. On the temp sensors this can lead to the controller reading incorrect resistance values which may not kill the engine; but, will cause poor operation. New or reproduction D jet connector plugs and the rubber boots are available if you need to re-do or re- terminate the plugs although they are kind of pricey.
Sometimes spray contact cleaners available from electronics suppliers can help improve the operation of a switch. I have found that the improvement tends to be temporary as the cleaners are really just degreasers and not effective at removing oxides on the contact surfaces. Best solution for rehabilitation is contact cleaner followed by burnishing the contacts and Oxgard on the contacts.
As a final note, your non starting when cold may be related to the ignition switch. The D jet gets its start signal off the coil + connection. If the coil is not getting +12v from the switch the D jet will also be dead.