More robust way of checking the browser type to fix Chrome 144+ issues

[Igloczek]

Overview

Introduce more robust way of checking browser type, based on variables set during the build, instead of the runtime checks, to address the problems reported in #2039

Manual Testing

console.log(browser.devtools)

shouldn't return undefined in Chrome 144+

Related Issue

#2039

This PR closes #2039

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[aklinker1]

@Igloczek looks like Chrome fixed this on their side: #2039 (comment).

I'm a little hesitant to make this change, it means people can't "reuse" a single extension build between multiple browsers and have it function the same way as before... but at the same time, I like this change. And I don't know how many people reuse builds between browsers (I would think very few, but not zero lol).

Can you rebase it off the major branch and target it in this PR? That way we can merge this into the next major release as a breaking change and ensure the change is documented.