I think I believe I feel statements should be introduced properly

This probably comes under the category of pragmatic linguistics. It is certainly very important in Neuro-linguistic programming (NLP).

People perceive the world via different modalities (vision, sound, feeling etc.).

In NLP it is asserted that a person's speech and writing will betray their dominant modality (modalities). Thus a person who is very visual (perhaps an artist) is likely to use visual terminology, e.g. instead of saying "I understand what you are saying" a visual person might say "I see what you mean", a musician might perhaps say "That sounds correct" and a sculptor could say, "I have a feeling you are right."

If you subscribe to this view then which a person chooses is personal to them. For that reason there is no objective way to compare such expressions except in the context of the individual who is using them.

Representational systems (NLP)

Representational systems (also known as sensory modalities and abbreviated to VAKOG or known as the 4-tuple) is a postulated model from neuro-linguistic programming regarding how the human mind supposedly processes and stores information. The central idea of this model is that experience is represented in the mind in sensorial terms ... Bandler and Grinder claim that each person has a "most highly valued" (now commonly termed preferred) representational system in which they are more able to vividly create an experience (in their mind) in terms of that representational system, tend to use that representational system more often than the others, and have more distinctions available in that representation system than the others.[4] So for example a person that most highly values their visual representation system is able to easily and vividly visualise things and has a tendency to do so in preference to creating or recreating an experience in terms of the other representational system.
Wkipedia

"I feel" means just that: something or someone has caused me to feel happy, or anxious, or resigned, etc.

"I believe" means that I accept as something as true without proof, be it your word, or religion, or a salesman's unwritten promises, or that I'll become a rock star.

"I think" means I apply rational processes to analyze information, its sources, its applications, its consequences, etc. From this process I might come to form an opinion. With sufficient evidence or proof, I might come to "know."

It makes me a bit nuts when people use these words interchangeablely; it lacks clarity. And if I wanted your feelings about something, I wouldn't have requested your thoughts.

"I reckon" means I calculate something. Perhaps I calculate a navigational course by means of dead reckoning, or reckon accounts. And in several regional dialects it is used synonymously with I think, I calculate, or I figure.