I'd like to have this new domain because I'd like to have "awesome" and "page" as keywords in the URL and not only in the name of the page.
Google puts little to no weight on keywords in the domain name at this point. Several years ago it appeared to be major ranking factor. Google changed their algorithms. Now it doesn't help and it may even hurt.
The main goal should be to be present, on google SERPs, with both URLs.
No matter how you do it, Google is only going to index your page under a single URL. Depending on how you host it, you can choose which URL Google indexes. If you have the content up on both URLs, Google will detect that it is duplicate and choose only one to put into the search results.
I'd like to have only the domain name and redirect the traffic on the original page
This sounds backwards from your chosen redirect solution. When implementing a "domain forwarding" redirect at the domain registrar, the new domain will redirect to your existing page. It sounds like you had wanted it the other way around.
On the redirect manager, I've been asked to choose between "keep the URL on the browser bar" or to redirect the user to the page with the address changed
"keep the URL on the browser bar" implements a "framed redirect" whereas not doing so will likely issue a 301 or 302 redirect. Framed redirects introduce problems when users click on links in the framed page. At that point the URL in the browser bar still doesn't change. If you are going to use a framed redirect you probably want to put a
<base target=_top> tag in the head of your page so that all links on the page break out of the frame set.
No matter what type of redirect you choose here, Google will continue to index your page at the original URL. Only people who type in your new domain name will be able to take advantage of it.
It sounds like you might actually want to host your page under the new domain name. As you note, you might have to buy more hosting for that. Moving a page to a new domain name is also likely to kill your Google rankings. Google has tools to help you move a whole site to a new domain, but doesn't support moving only a page.
Do I need to add this new domain to GWT?
If you are only redirecting the domain, there is no great benefit to having it in Google Search Console.