Don't get too precious over the term "resolution". There are many ways to define it, and indeed ultimately what you resolve with a microscope gets down to what measurement signal to noise ratio you can achieve. With a perfectly clean signal, you can deconvolve the lens's point spread function from your image and resolve features smaller than what the simple formulas imply. The "diffraction limit" is not a hard limit since it is a lowpass spatial filtering: you can reverse the lowpass by deconvolution if the noise levels allows. Practically, though, you can seldom do this. Often when you work out the number of photons per second coming from each resolvable volume in microscopy, it's surprisingly low and hence the quantum limit is going to hit you.
The first formula is found by measuring the diameter of the first zero in the perfect, unapodised point spread function ("Airy Disk") given by $\frac{J_1(k\,\eta\,r)}{k\,\eta\,r}$, where $\eta$ is the numerical aperture. The Bessel function $J_1$ has its first zero at 3.83, hence the "resolution" measured this way is $2\times 3.83/(\eta\,k)$, which yields your first formula if you multiply it out (since $2\times 3.83 / 2\pi = 1.22$).
Other ways of defining resolution are by quoting the point spread function's full width half maximum. Still others quote diameter that encircles some fraction of the power through the diffraction limited spot, and this fraction can vary: $95\%$, $1-e^{-1}$ and $1-e^{-2}$ are all common.