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That just transfers funds to your savings. You have to wait 3 days to get your money out of savings I believe. There isn't an option to convert into steem but you can just go to the "market" and buy some steem there with your sbd.

My guess was: I have SP, SDB and STEEM on my wallet, so I thought my savings were steem (if not where are they supposed to go now?)
It's weird if there is no way to power up without bringing money outside the platform, isn't it?

There is. I haven't put any money into Steemit and I powered up 40 Steem. You just need to click on the steem amount on your wallet and choose to power up. Savings are only used to secure your funds in case of hack for example. It would take 3 days for the hacker to get out your money.