It always puzzles me when websites say we should bury bulbs under the soil about 2 or 3 times the height of the bulb.
For example, a nice post about planting hyacinths recommends putting the bulb 5 cm beneath the soil, while all the planted mature, ready to flower, hyacinths I've seen on the market have about two-thirds of the bulb above the soil.
If I've seen the bulb growing a bud, should I bury the bulb?
Yes, and as recommended by the supplier. The ones you see in bowls, ready to flower, are for indoor use and have been specially treated to induce them to grow for use as houseplants. They are also usually planted in bowls without drainage, in a special growing medium, and they are not expected to be kept for long, because once the flowering is over, they won't flower again in the house, and need either to be binned or planted outside, at the recommended depth of 2-3 times the depth of the bulb itself.
Yes, the supplier's recommended depth means that there is enough soil on top of the bulb to protect it during the winter and keep it from emerging too early. You should follow the recommendation. This means that if you have a bulb that is 5 centimeters tall, you should have 10-15 cm of soil on top of it.
That said, I have planted some very hardy bulbs shallower than recommended with great results. We have severe winters where I live, but daffodils seem to do especially well here. I've planted some bulbs just 5 cm under the soil, with the understanding that they may not naturalize as well, but if they survive, I'll have flowers at least a week earlier than I would have otherwise. The bulbs I plant shallowly like this are always the first ones to come up in my neighborhood by far and are very welcome after the long winter. It's worth the risk of losing them over the winter. I'd only do this with very hearty flowers, though.
As for your bulbs that are sprouting, there is no harm in planting them, but they may not flower in the spring.
