First thanks for the tip @stephcurry

You made a small mistake in your tutorial, it's actually on the "DEPOSIT" button that you want to click, you can then select the "Coinbase Account" where the the text says "Select a Coinbase wallet and the amount you wish to transfer TO your GDAX trading account".

As explained stephcurry:
You ONLY get free conversion from fiat to crypto (USD to BTC in this case) on GDAX only if you are a "maker".
That means if you ask for a price below the currently listed price (your trade request stay "stuck" until that price is matched).

Sources:

• GDAX fee page on gdax.com/fees/BTC-EUR
• "Maker taker fees in plain english" on money.stackexchange.com/questions/63044/maker-taker-fees-can-somebody-explain-in-plain-english

More about GDAX

GDAX belongs to CoinBase, it's just "one of the 2 products" of CoinBase Inc.

GDAX acronym stands for Global Digital Asset Exchange.

It's the kind of pro trader version of CoinBase exchange.

It's backed by the New York Stock Exchange & venture capitalists like Andreesen Horowitz.

It was the first regulated crypto exchange in the US (since 2015).

"All digital assets held in online storage are fully insured" they claim.

"98% of customer digital assets are stored entirely offline” they claim.

Preety huge volume: in Sept 2017 "day trade volume record of 650 millions USD across BTC, ETH & LTE" twitter.com/GDAX/status/908757856513441793

See also en.wikipedia.org/wiki/Coinbase

So yes, it does seem like a very decent platform and company behind it.

But as always, don't trust everything you read from unknown individual (me!) so do your homework before using any exchange. I'm pretty sure stephcurry will agree with that.