Are there any diagrams or tables of relationships like with groups to magmas, but for rings or fields?

For groups we have these:

Does anything like this exist for rings or fields in abstract algebra?

I see this:

But I am not sure if we can go any deeper in demonstrating the relationships between groups/rings/fields and other "intermediate" sort of forms. If no diagrams/tables exist, what properties would need to be collected in order to build such a table from scratch? If no such similar relationships can be drawn for rings or fields (like perhaps "ringoids" or "quasirings"), well that would be good to know too. I am new to this.

The collection of species of ring-like objects with names does not really lend itself well to a taxonomy like this.

There are three sorts of "dimensions" you can vary the definition of a ring with:

1. alter the $+$ group operation
2. alter the $\cdot$ monoid operation
3. alter the distributivity condition

Here are the major ones worth knowing:

$+$ group modification

If you relax $+$ to be simply a commutative monoid, this is called a semiring

$\cdot$ monoid modification

You could take away the requirement for an identity from the monoid, in which case you'd be talking about a ring possibly without identity.

If you make the monoid commutative you'd be talking about a commutative ring

If you made the nonzero elements form a group, you'd be talking about a division ring.

If you dropped associativity from the monoid you'd have a nonassociative ring, of which there are many interesting species

distributivity modification

If you only require $\cdot$ to be distributive over $+$ on one side this is called a nearring.

Now, there are of course many ways to combine these and the possibilities stack up fast. Really I don't think there are very standard names for these mixtures in common use. There are a few books that attempt taxonomies, and the ones I'll point you to are

Gondran, M., & Minoux, M. (2008). Graphs, dioids and semirings: new models and algorithms (Vol. 41). Springer Science & Business Media.

Golan, Jonathan S. Semirings and their Applications. Springer Science & Business Media, 2013.

Now, somewhat more interesting is looking closer at the "dimension" of varying the associativity-type of the monoid.

If you take a look at this wiki you can build an interesting graph between properties between the conditions in this set: {associative, alternative, flexible, Jordan}.

To me the most important categories of not-associative rings are Lie rings and Jordan rings.

I feel like if you want to put "ring" and "field" on the same chart, then it makes a lot more sense to put lots of interesting ring-theoretic conditions on the same chart, and just have "ring" be the one at the top. That's what I did when I generated these charts.