I prefer not to damage the potato with tooth picks. They can start to mold and rot from the scars.
I also had better success laying the tubers on their sides so full length of the one side of the tuber is moistened -- reason for this is sometimes it's difficult to determine which way is the tip end and which end was attached, and also, the tubers seem to grow shoots/slips in clusters and it's hard to tell where they are going to grow from.
If you suspend one half in the water you might end up with the critical end submerged.
They like to be 80-85Â° to start growing. I have trouble getting them started until it's warmer in spring -- starting earlier just means watching the tubers do nothing until we start getting those sunny sudden heat waves (Typically not until April here) -- but if you can get them growing by about a month before planting (soil needs to be warm, too so at least not until your garden is warm enough to plant beans and melons) you'll be fine. Once they start to grow, they are monsters and will quickly take over the (indoor) space.
I think about two weeks for the sprouts to grow enough to slip size, two more weeks to remove and grow the slips individually with own roots, then plant in garden after the soil is warm enough is approximately the right schedule. You don't want the slips to be too overgrown and still have to keep them inside. So specifically, 90 days seems too far ahead to me?
I also had success growing them by half burying them in sandy mix in a container and double bagging in black and clear plastic bags, then keeping them in sunniest part of the garden (this was obviously after the last frost).