Ahhh.

Well essentially, Hive was an exact copy-paste of Steem, 5 weeks ago (or so), when shit went down. Somebody pressed a magic button, and there was a replica, with a new name, new URL, etc.
Accounts were also copied. Wallets too. And passwords. Basically everything is the same.
But from that moment on, changes made to one didn't change the other. They are separate, and moving apart slowly. Even the price of the 2 currencies are, as you noted, not moving together - any more than any other cryptos are.

So, for moving funds around...

I'm unbanked, and while not a criminal (whatever that means anyway), I don't like attaching my personal identity to every transaction I make. I feel that gives somebody(s) too much information about me, and info is power nowadays. I don't like giving away my personal power. I think you're similar, so I imagine you'd rather keep your banking info separate from your cryptos. I think using cryptos is about decentralization and some degree of privacy. I feel that people give up the privacy side when they attach their government ID or bank accounts to their cryptos.

Long story short, I don't use big exchanges that require identification.

So what I did and am still doing with my STEEM is converting it to HIVE through this process. Note, there may be better ways to do it. I'm definitely still learning all this stuff myself even though I've been following Bitcoin since 2011 :P

1. Power down the STEEM.
2. Log in to steem-engine.com with your posting key.
3. If this is your first time, get a hang of the site, how to use the market screen (for converting between tokens which requires your active key), etc.
4. Use the "deposit" button to pull your STEEM into your steem-engine wallet. They become Steem-backed tokens there (I assume for regulatory reasons or something).
5. Use the market screen to buy Hive-backed tokens with your Steem-backed tokens. A bit of strategy is involved here, and I'll let you handle that. Either pay whatever is the best going rate (faster), or throw up an offer for whatever price suits you, and wait for it to be filled (slower).
6. Back in your wallet again, use the "withdraw" button to send the HIVE to your wallet on the Hive blockchain.
7. Takes about 1 minute to be available. Power it up to HP, transfer it to me, and/or keep it liquid for later, etc. ;P

So yeah, it's probably fairly easy if you've used steem-engine before, and probably still not that tough if you haven't. But if you haven't, cool, there should be a bunch of various tokens in your wallet there, waiting for you. Some will need to be claimed using the "rewards" option in the pulldown menu, but most should already be sitting in the wallet. When I first checked, I had like $30 worth of various tokens. They're earned for doing all kinds of things on the blockchain, given as awards, gifts, handed out randomly like airdrops, and so on. I can't see your wallet (they're private, unlike wallets on the blockchains), but I imagine there will be various things in there. I've sold most/all of mine for more Hive, but some people think tokens are good to hodl. That'd be up to you!

Oh, as for security. I haven't had a problem with that site, and haven't heard of anyone who has. It's a bit amateur in some ways, "unprofessional" perhaps some would say? But it suits me because it works. I like functional shit. They do take a 1% fee on deposits and withdrawals, but all trading is free, so you can convert between tokens all day and night without paying anything. So if you're just pulling funds in, doing some converting, and pulling them back out afterward, you'll pay a total of 2% in fees. Just comes out of the payments.

It's kinda hard to do anything wrong, aside from maybe making shitty trades, but you'll know if you're doing that. You can't just hit a wrong button and lose everything, or anything like that. Like I said, there are probably many other ways, but this way suited me because I didn't have to sign up for anything, I didn't have to sacrifice my privacy, it didn't cost much, it was instant (aside from the learning curve), and it worked properly. If you value similar things, it might be an avenue to check out.

Anyhow please let me know if any steps weren't detailed enough. I'm unsure of your level of expertise with blockchains and cryptos. Enjoy the rest of your weekend : )