Yuval's birthday attack I found this paper:
https://www.researchgate.net/profile/Ganesh-Gupta-7/publication/271704029_What_is_Birthday_attack/links/54cfbdcc0cf24601c0958a1e/What-is-Birthday-attack.pdf
The following attack is outlined on page 9:
Input
-----
Two types of messages - legitimate message 1 ; fraudulent message 2 ; m bit
length; one -way hash function H

Output:
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1′, 2′ Is a minor modification of 1, 2 with (1′) = (2′).

1) Generate  = 2^(/2) minor modifications of 1′of 1.
2) Hash each such modified message, and store the hash-values such that they
can be subsequently searched on hash -values. This can be done in () total
time using conventional hashing.
3) Generate minor modifications 2′  2 , computing (2′) for each and checking 
for any matches with any 1′ above; continue until a match is found.

I'm confused about step 3.  How would (1′) = (2′)?  I thought the birthday problem meant finding collisions such that (1) = (1′), and (2) = (2′)?
Yuval's attack is slightly different from the standard birthday attack where we look for a repeated output in a single family of inputs. Instead we look for a repeated output across two families of inputs with at least one member of each family producing the repeated ouput. The probabilities are slightly different, but in a complexity sense are both $O(\sqrt H)$ for output ranges of size $H$ (or $O(2^{m/2})$ for $m$-bit output values) where in the worst case outputs are approximately uniformly distributed.
One can see this in an approximate sense as generating $n$-outputs in each family creates $n^2$ pairs of values with one half of each pair coming from each family. Each pair has a $1/H$ chance of matching the two halves. We can approximate the chance of finding a repeat as
$$1-\left(1-\frac1H\right)^{n^2}$$
and for $n\sim\sqrt H$ this will be about $(1-1/e)$. Note however that this is not a rigorous estimate as the probabilities are not independent. More rigorous derivations are possible (see Girault et al., Eurocrypt '88).