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I agree, these are somewhat dubious designations, but there's a possible justification for looking at them more or less as analyzed in your example, in increasing order of dubiousness.

The first example is actually just a deceptive resolution of the secondary dominant, akin to a primary V7 going to vi. V7/vi in C major is an E7 chord that wants to go to A minor—that would be the i if we were transposing to the key of the vi chord. Instead the chord resolves to F major, which would be the deceptive resolution to VI in the same key, or VI/vi in the actual C Major key. F major and A minor triads share two notes, A and C, and this is part of why any deceptive progression works, two of the three notes in the target chord are the same.

The third example is a bit funkier, but it's worth noting that the target chord of the V7/IV does come, it's just delayed by the bIII on the way. Delaying the resolution of a secondary dominant with a brief interjection is actually pretty common, although bIII is a fairly uncommon chord to throw into the mix. It essentially functions like a bVII/IV, which is a fairly common resolution in rock/pop/jazz harmony. So I think I might call the progression I — V7/IV — bVII/IV — IV, etc.

I'm still thinking about the second example. I'll edit my answer after I've thought about it a bit more.

EDIT: OK, having meditated on it a bit, I find the second example to be the least effectively understood as having anything to do with secondary dominants. I hear this more as a really nice example of linear harmony—parsimonious connections between various triadic structures largely by means of stepwise connections combined with common tones. The move from C to D7 is glued together by the common tone C. The move from D7 to F is particularly gorgeous, and—if we temporarily ignore the seventh of the D chord—is common enough to 19th-century music that it has a name, the so-called "chromatic mediant." Any time you have two major chords (or two minor chords) that have roots a third apart, you have a chromatic mediant relationship. D major to F major is definitely an example. There will always be: 1) a common tone, in our example A; 2) a chromatic modification, in our example F# moving to F natural; and 3) a stepwise movement, in our example the D moving to C. Moving between chords that are chromatic mediants thus has a relatively close connection in that all the notes are either the same or a step away, but they also have a freshness to them because the two chords never could belong to the same key. It's one of my favorite tonal connections. The seventh in the D7 chord in this particular example just provides further glue in that the C is a common tone with the preceding chord AND with the following chord. In terms of a Roman numeral analysis, I would not try to make up numerals for the D7, I'd just call it linear harmony that gets to IV via the chromatic mediant relationship, and then moves plagally to I. Does any of that make sense? I can edit for clarity if not.

I agree, these are somewhat dubious designations, but there's a possible justification for looking at them more or less as analyzed in your example, in increasing order of dubiousness.

The first example is actually just a deceptive resolution of the secondary dominant, akin to a primary V7 going to vi. V7/vi in C major is an E7 chord that wants to go to A minor—that would be the i if we were transposing to the key of the vi chord. Instead the chord resolves to F major, which would be the deceptive resolution to VI in the same key, or VI/vi in the actual C Major key. F major and A minor triads share two notes, A and C, and this is part of why any deceptive progression works, two of the three notes in the target chord are the same.

The third example is a bit funkier, but it's worth noting that the target chord of the V7/IV does come, it's just delayed by the bIII on the way. Delaying the resolution of a secondary dominant with a brief interjection is actually pretty common, although bIII is a fairly uncommon chord to throw into the mix. It essentially functions like a bVII/IV, which is a fairly common resolution in rock/pop/jazz harmony. So I think I might call the progression I — V7/IV — bVII/IV — IV, etc.

I'm still thinking about the second example. I'll edit my answer after I've thought about it a bit more.

I agree, these are somewhat dubious designations, but there's a possible justification for looking at them more or less as analyzed in your example, in increasing order of dubiousness.

The first example is actually just a deceptive resolution of the secondary dominant, akin to a primary V7 going to vi. V7/vi in C major is an E7 chord that wants to go to A minor—that would be the i if we were transposing to the key of the vi chord. Instead the chord resolves to F major, which would be the deceptive resolution to VI in the same key, or VI/vi in the actual C Major key. F major and A minor triads share two notes, A and C, and this is part of why any deceptive progression works, two of the three notes in the target chord are the same.

The third example is a bit funkier, but it's worth noting that the target chord of the V7/IV does come, it's just delayed by the bIII on the way. Delaying the resolution of a secondary dominant with a brief interjection is actually pretty common, although bIII is a fairly uncommon chord to throw into the mix. It essentially functions like a bVII/IV, which is a fairly common resolution in rock/pop/jazz harmony. So I think I might call the progression I — V7/IV — bVII/IV — IV, etc.

I'm still thinking about the second example. I'll edit my answer after I've thought about it a bit more.

EDIT: OK, having meditated on it a bit, I find the second example to be the least effectively understood as having anything to do with secondary dominants. I hear this more as a really nice example of linear harmony—parsimonious connections between various triadic structures largely by means of stepwise connections combined with common tones. The move from C to D7 is glued together by the common tone C. The move from D7 to F is particularly gorgeous, and—if we temporarily ignore the seventh of the D chord—is common enough to 19th-century music that it has a name, the so-called "chromatic mediant." Any time you have two major chords (or two minor chords) that have roots a third apart, you have a chromatic mediant relationship. D major to F major is definitely an example. There will always be: 1) a common tone, in our example A; 2) a chromatic modification, in our example F# moving to F natural; and 3) a stepwise movement, in our example the D moving to C. Moving between chords that are chromatic mediants thus has a relatively close connection in that all the notes are either the same or a step away, but they also have a freshness to them because the two chords never could belong to the same key. It's one of my favorite tonal connections. The seventh in the D7 chord in this particular example just provides further glue in that the C is a common tone with the preceding chord AND with the following chord. In terms of a Roman numeral analysis, I would not try to make up numerals for the D7, I'd just call it linear harmony that gets to IV via the chromatic mediant relationship, and then moves plagally to I. Does any of that make sense? I can edit for clarity if not.

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I agree, these are somewhat dubious designations, but there's a possible justification for looking at them more or less as analyzed in your example, in increasing order of dubiousness.

The first example is actually just a deceptive resolution of the secondary dominant, akin to a primary V7 going to vi. V7/vi in C major is an E7 chord that wants to go to A minor—that would be the i if we were transposing to the key of the vi chord. Instead the chord resolves to F major, which would be the deceptive resolution to VI in the same key, or VI/vi in the actual C Major key. F major and A minor triads share two notes, A and C, and this is part of why any deceptive progression works, two of the three notes in the target chord are the same.

The third example is a bit funkier, but it's worth noting that the target chord of the V7/IV does come, it's just delayed by the bIII on the way. Delaying the resolution of a secondary dominant with a brief interjection is actually pretty common, although bIII is a fairly uncommon chord to throw into the mix. It essentially functions like a bVII/IV, which is a fairly common resolution in rock/pop/jazz harmony. So I think I might call the progression I — V7/IV — bVII/IV — IV, etc.

I'm still thinking about the second example. I'll edit my answer after I've thought about it a bit more.