Looks like 'Irene Nuss'. Younger leaves on new stems often are silver-streaked. The bigger leaves are usually all green - often dark green outdoors. Not sure what you are referring to "are these buds going to grown into new stems".

Irene is an old hybrid and named after another grower. It is a superba type cane. An oldie but goodie!

Younger stems might look like this to begin with.

Tons of pink flowers if conditions are right.

Nice thick big leaves from this one that was "winter pruned to the ground by Jack Frost a few years ago".

Red reverse leaves.

Full sun often brings on lots of blooms but dappled shade can have similar results.

I know of no way to keep the silver splash in the younger leaves though. Just enjoy it for what it does.

Choose some other varieties like 'Silver Mist' or 'Lana' for more silver.

Here is my Silver Mist from last year (grown from cuttings in one year). It has lighter pink blooms when it gets to the right age/conditions.

'Lana' seems to keep its silver streaks most of the time and is an easy bloomer (not blooming in this photo though)

Your last photo is probably some garden variety of coccinea which are much hardier and easy bloomers - usually pink.

Thank you for you answer. I thought initially it was a Sophie Cecile.

Your begonias are beautiful!

My photo of the buds was not very clear. I took another one. Are these going to be bloom stalks or stems?

those are potential stems. pinch the top out to force growth at the nodes

Thank you! I will do that. I would like this begonia to look a little bit bushier.

Good luck with your plant