How do I approach learning Data Science/ML the 'rightest' way?
Welcome to the community!

I normally don't answer questions like this, they're pretty broad and there are many others just like it. It seems like you're not actually asking how to get started, you already seem set on reading some books (which is great!), so let me answer that question, rather than a very broad 'how to get started'.

First of all, I'd like to challenge your assumption about learning from the bottom up, and offer an alternative opinion (yes, this part is just an opinion). I'd instead encourage you to get your hands on the high level tools and start using them right away. If everything we ever did required us to forego abstractions and learn everything from the beginning, we'd still be reinventing the wheel over and over. Start experimenting with the existing tools and supplement that with your reading. You'll develop an excellent understanding this way.

So to answer your question... I'd recommend the second book you quoted: Hands-On ML with Scikit Learn and Tensorflow. This book will teach you good habits right from the beginning, give you a good understanding, and get you using those tools right away.

Best of luck.