There are two aspects to this:

(1) Appearance: icon only vs. icon + name (2) Behavior: how you interact with the search engines

I can help you change #1, and I can explain how to work most efficiently with #2, but there's no way to change #2. Here we go.

Using the Bar

Seeing your current default search engine

OLD BAR: icon in the search bar NEW BAR: listed at top of drop-down

To use this search engine, enter your query and press Enter or click the arrow. No need to click anything on the drop-down.

What if you want to send your search to a different site and make that site your default search engine?

OLD BAR: click the new default site (or arrow down to it and press Enter) NEW BAR: right-click the new default site, Set As Default Search Engine (or Ctrl+arrow to cycle through them and press Enter)

Then enter your query and press Enter or click the arrow.

What if you want to send your search to a different site WITHOUT making that site your default search engine (one-time search)?

OLD BAR: not possible NEW BAR: enter your query first, then click the button for the site (or arrow down to it and press Enter)

Bar Appearance

Many parts of Firefox's UI can be restyled using custom style rules. You can apply those rules by creating a userChrome.css file. Allow 10 minutes for this.

To generate custom style rules for the search bar

I have a style recipe generator here:

https://www.jeffersonscher.com/gm/search-bar-names.html

Using those rules in a userChrome.css file

I have a website with more information: https://www.userchrome.org/

I recorded a comparison video: Firefox Old Search Bar vs. Restyled New Bar.

Following your video after I create the shortcut, FF is opening the CSS file as text. Nothing happens.

derloopkat said

Following your video after I create the shortcut, FF is opening the CSS file as text. Nothing happens.

Is that at the point where the CSS file needs to be saved to disk? Try right-clicking the link and using Save Link As if you do not get a download dialog automatically.

It's what I did. I have downloaded the file, renamed, moved to profile folder into userChrome folder, The shortcut opens CSS file as text.

Let's ignore the shortcut for a moment.

You have your semi-randomly-named profile folder (for example):

123abc4d.default

In that folder you have a folder named

chrome

In the chrome folder you have the downloaded file that you renamed to

userChrome.css

With that arrangement, when you quit Firefox and start it up again, Firefox should find the file and apply it to the search bar.