Good job! (and welcome back, again.) Yes, 5x4 is a pretty difficult very low-res design (less tough than e.g. 4x5 and 4x4, cf. AGH.) Keep up your interesting experiments and usable fonts.

Well done! (it's not my criterion, but Unicode's b.t.w. - we need standards, else 'fonts' are personal and not quite usable.)

So, after reading there, I complete by detailing even more (since you tend to specialize into minimal pixel fonts, fine!).

Your FLAWS of design:

• Lowercase 5x4 grid : it collides the height of the grid reserved -by priority- to the uppercase (5x4), which makes e.g. your lowercase "p" behave as a uppercase "P" (which is more serious than an ambiguity; gqyz have almost the same issue, which can be fixed too, I showed you in a tougher design);

• Grave accent : FS hack or not, the sad fact is that your "fs 5x4" is actually 5x5 by doing so (you even ask the user to fix this glyph by cloning it, I assume, which makes your '5x4' design quite unusable as is in US-ASCII);

• You could not design a strict 4x4 set of numerics (quite fixable "1" too);

• We can all see how you designed this font: likely a strict 5x4 uppercase first, next the 4x4 numerics (incomplete job for a strict sub-grid), the quite loose (mixed grids) lowercase (variable x-height which is unserious), and the punctuations/symbols with no limitation of grid. Again, you cannot compare a strict and hard 5x5 (in 3 out of 4 US-ASCII subsets) without your creation that took you only a few minutes of fun, even though I agree that stretching everywhere to 5x5 is not recommended for any context (in pixel art, the even look might help in some situations). This point is not a direct flaw, but if one considers the incoherent subgrids, we can spot many errors;

• '" and "' are unusable for the human (' duplicates "), and -I repeat this here since 2013- this monospace hack is poor (see e.g. this link) and should be avoided (even if we'd prefer these punctuations straight).

So, what I'll recommend you (to try - not exhaustively):

unchanged 5x4 uppercase (even though I already indicated a few ambiguities - but you still need to learn pixel art, see. e.g. "P" v.s. "R");

+ 4x4 lowercase (so that you don't waste the grid anymore and can add your 4 x-height ex-hacks)

+ 3x4 numerics (not 4x4 any longer because of the latter suggestion)

+ (almost?) unchanged punctuations/symbols (1x1 till 5x4, and not 5x5 of course, and if you need your font to be compatible as proportional, edit ' or/and ").

In my new suggestion above, I left several inconsistencies from your latest update (i.e. potential ambiguities).
I remind you of the problem: if you set a rigid rule of square or round glyphs, you lose many solutions (similarities being chosen over the discriminations).
As a result, many difficulties might spoil the readability, pixel-wise.
This is the same syndrom as wanting to put a black square in any ambiguous situation (i.e. unmastered compression = too much weight, or/and overlaid and unmodified segments of glyphs).
But in my opinion, your "fs 5x4" confirms the lack of conventional pixel art tools in the current fontstructor, especially an overview of all the grids in a strict and visual grid.
I mean, you repeated the same ambiguities, without fixing them, even without understanding why I edited your flawed glyphs (you can check and compare after the list below).
Let me give you these examples of the lack of concern in your design (if I were you, I'd use a conventional bitmap editor before any of such low-res fontstruction):
• C < G (+ 1 dot)
• K < H (+ 1 dot)
• O < Q (+ 1 dot)
• a < d (+ 1 dot)
• 6 < b (+ 1 dot)
• B ~ D (1 dot moved)
• c ~ o (1 dot moved)
• J & Ss (uncommon shapes)
• ? v.s. ! (unpaired)
• _ (n/a shortening)
• |k ~~ |< (similar)
• & ~~ &: (ambiguous)
• jftw (incomplete compression)
• non US-ASCII (providing 8-bit punctuations (quotes) is a deceptive feature in the fontstructor, I assume, so please refer to Unicode instead).
Etc.
Oh, and please bear in mind to not mixing the designs with glyphs that have the full width (5-dot here), and -a lot- simpler ones that are narrower into the main matrix, even variable/messy (like your preference here).
The best-looking monospace fonts use glyphs that have an homogeneous width.
If you need to compare my monospace 5x4 edition, with your 5x5 'monospace' 'font' (filled with proportional and easier shapes), then you might still need to learn about their contexts of use, I think (e.g. there are many cases beyond the simple differentiation of 0 v.s. O).

BOTTOM LINES:
1. your fonts are not strict US-ASCII (they have and artistic value, but none in e.g. coding as they are);
2. you introduce too many ambiguities (this list above, plus the abuse of horizontal bars - the square rule you might have added to the uppercase);
3. 5-dot of width have so easy-to-do designs that I did not even add any to my main projects! (4x5 max for the comparison, but 3x5 max actually.) Please remember that 4-dot of width is the great maximum we can afford in very low-res fonts (else anyone, even the blind can draw such wide glyphs);
4. as always, one can free oneself from my experiments that employ the full widths (I fell sorry you did not get this), e.g. you can edit the 5x4 lowercase i into a 1x4 if you need/want (the context decides, ultimately).

“Sur ce, excellentes fêtes de Noël à toi !”
= Merry Xmas!

Examples of unreadability & illegibility…