The Merida Font
As noted by Tardigrade, the Merida chess font is available from https://github.com/vasiliyaltunin/chess-merida-font. However this is a derived work which incorporates a much older file containing the actual glyphs in TTF format.
Edit: If you google for "Armando Hernandez Marroquin" you will find a lot of font download sites offering several chess fonts by Marroquin. I've sampled a couple of the README files from these fonts, both of which contained the same freeware statement quoted below.
The source/README_D.TXT file referenced from the front page seems to be by the original lead author Marroquin speaking on behalf of the other authors, and says:
The font
is freeware, I hope it is useful for the chess buddies. If you want to repay
us, you can send us some other fonts for creating chess diagrams (we are
collecting them).
As far as I can see the TTF font file by Marroquin et al has been incorporated directly into this repository, so if you were to use that alone then the original "freeware" grant would apply. The Mozilla Public License only restricts the other works in the repository (contrary to what the repository README claims). TTF is a compiled form of the font, so that would explain the use of the term "freeware": this is not open source.
It is true that "freeware" is not a formal license. The usual definition is "software made available free of charge". The following note about "If you want to repay us" would also seem to indicate an intent to make it available to all without requiring a fee.
The Marroquin README also says
The figures of this font follow the more traditional style of the figures found in many publications with chess diagrams. I took them from a book with more than 5,000 chess problems.
This means that Marroquin et al probably cannot claim any IPR in the visual appearance of the font glyphs. It is possible that some expressive content might be found in the exact proportions of the various glyphs, but to the extent that they merely copied traditional figures into a computer there was no creativity and hence no copyright.
However Marroquin et al could claim copyright in the resulting TTF file, and any other format derived from it which describes the font in terms of vector shapes. The appearance of a piece does not dictate any single representation in geometric form in a font glyph, so the decisions made by Marroquin et al were creative. The only exception would be if the diagrams were represented using bitmaps, as that would merely be a representation of the appearance and not include Marroquin's creative work.
This matters because it affects electronic and paper publication differently:
If your book is published in paper form then Marroquin et al cannot claim copyright to the visual appearance shown on the page.
If your book is published electronically in a form that embeds the TTF file or some similar format derived from it then Marroquin's copyright would apply to that file.
If your e-book only contains bitmap versions of the chess piece glyphs then you are in the clear for the same reason as the paper version.
Whether a publisher's legal department would consider this sufficient is another matter.
Chess Diagram Editors
The Apronos.com website you reference says that it is free for non-commercial amateur use only. However this is a restriction on the use of the website itself; if you use it to create diagrams which you subsequently publish then the website owner has no rights to your work, only to compensation for violating his terms of service (such as they are), and to an injunction stopping you from doing it again.
If the website offered a commercial paid service then they could claim the damage as being the amount you avoided paying. Without such an alternative they could only claim the amount that you cost them by using the service, which is going to be trivial.
If you want to be absolutely safe you can always email the website owner and ask if you can have permission in return for a contribution to running costs.
