I initially put heel lifts in. Ran on grass. That was lame and lasted about a week. I then wore normal trainers. I then very slowly went from trainers to more minimal footwear, BUT I pretty well avoided flat heel-to-toe offsets. Maybe your Achilles problem was a freak tear/pull/stretch, but more than likely you have a lifetime of wearing built-up shoes that you'll have a hell of a time reversing now that you're beyond puberty.
If you run very conservatively and take a lot of time to recover between running attempts and actual runs, then you'll probably get away with doing either. I think your best bet is to get ample heel lift until the Achilles has time to heal. Then test how well you can stretch/lengthen things.
Lots of people will disagree because they've done it a certain way. My personal experience is that going minimal was fine, so long as I still wore shoes with a decent heel offset.
I thought the Cumulus was soft when I wore it. I think soft shoes are bad. They let your foot sink and do things in one direction when your gait is trying to take you another direction. Better to have a flexible shoe without too soft of an insole. And then just run in grass while you're recovering.
Look for shoes with a comfortable heel counter or lack thereof. Even the wrong rub on the heel can irritate the Achilles enough that it doesn't heal as quickly.
Pain is bad. Don't do anything if you have pain or can't get your Achilles/calves limber. I would go out every day and try to limber up and jog. If I didn't feel good, I would stop. It's like that for a week or two, but after a few weeks you will probably find it okay to do some jogging and then things start to come back very quickly after that, but this will bother you for a year or forever. A chain is only as strong as its weakest link.
Warmth promotes bloodflow. Cold slows inflammation. You'll probably need both. Warmth is better before exercising and cold is better after. Hot and freezing are too extreme. Massaging is probably better than stretching, but both are good when done lightly. I hesitate to say any of this, because it's really easy to overdo any of these things.
Stay hydrated with an even mix of electrolytes and such.
In conclusion, I'm rambling. But I recommend keeping your heels slightly elevated until the heel isn't the problem. Once your knees or something else hurt more, you've made progress! It's a balance. The important thing is that the Achilles needs bloodflow to heal, so just sitting around isn't doing anything to heal it.