power analysis/ sample size

#1
Dear reader,

I would like to conduct a clinical study in which I compare 2 treatments. The goal is to determine which one cures the lesion faster. There are currently no studies comparing these two treatments head to head but there are studies comparing them to placebo. According to the literature these are the estimated durations for the lesions to cure:
- mean (in days) of golden standard 7.0 (according to the literature)
- mean (in days) of intervention 5.0 (according to the literature)
- SD 2.25 (according to the literature)
- We consider that 1 day is clinically relevant difference.
- alfa 0.05, beta 0.2

According to one of my supervisor this is the formula:
n = [2 x SD² / (7 – 6)² ] x ʃ(ά.ß) = [2 x 2,25² / (1)² ] x 7,9 = [2 x 5,0 / 1 ] x 7,9 = 10 x 7,9 = 79

The '6' is 7 minus the 1 day that is considered clinically relevant
However, I found on many sites that it is actually minus the expected difference. This would mean it would be:
n = [2 x SD² / (7 – 5)² ] x ʃ(ά.ß) = [2 x 2,25² / (2)² ] x 7,9 = [2 x 5,0 / 4 ] x 7,9 = 10 x 7,9 = 19.75

The difference is considerable. I understand that one needs more patients to determine a smal difference. Am I correct in thinking that if I expect a bigger difference that the sample size needs to be smaller despite the fact that a clinically relevant difference is smaller?

thanks in advance !