What's the real difference between a postdoc and a research fellow? (Which one is considered a 'better' experience?'
Q: It's not clear to me the actual difference between a postdoc and research fellow, especially when looking at the typical academia road.
Correct. Often times there is not a difference.
Titles are institution specific (e.g., universities, national labs, funding agencies) and often sub-division within institutions (e.g., departments, even specific PIs based upon how they hire people with their institution's hiring mechanism).
Sometimes a title including post-doc or similar title (e.g., post-doctoral fellow) requires the person be within X-years of their terminal degree.
Conversely, as noted by others, research fellows or similar title can be permanent positions, be programs or even titles for experienced scientists, or be visiting programs for distinguished faculty and researchers from other places.
For example The American Association for the Advancement of Science (AAAS) has a Fellowship Program for people (ranging from recent grads to tenured faculty) to gain experience applying science to policy.
You ask about demonstrating Qualities like 'Independent researcher that can design and lead high quality research programs'.
Rather than looking at title, demonstrate these qualities through outcomes.
For example, your CV might include (order the text based upon your CV's target such as a research or teaching school):

Position title, Place Name, supervised by Dr. Smith (2019 -2020)
Developed new widgets theory resulting X publications (depending upon your CV format, perhaps cross-ref here, e.g., Jones et al. 2019, Smith et al. 2020). Wrote grants for $XX,000 and managed the widget research project. Mentored Y peons. Presented results at XYZ. Developed and taught new course, Widget-ology. List other relevant outcomes to the position you're applying for.

Of course, list your grants, presentations, and publications in their own, respective sections of your CV.
Depending upon your subfield, the Place Name and the PI you worked under can be given a lot of weight or little weight compared to your outcomes (e.g., pubs, grants, technical skills).
Sometimes Position Title can carry weight, but these are usual for hard to obtain fellowships that are also captured in other parts of your CV (e.g., a US NSF Postdoc Fellowship that you obtains as the PI would also be listed under Grants on your CV).
Compared to a position's nouns (e.g., titles, places, PIs), actions usually carry more weight.
For example, publishing a first authored paper in Science or Nature would outweigh where you had your post-doc fellowship.