Since MyHeritage has partnered with Family Tree DNA, that is where the testing will be done, I imagine.
(FTDNA has sent out a reverse offer, a deep discount on a MyHeritage membership for FTDNA users.)
Many of us recommend FTDNA as the best place for genetic testing, in large part because of the matching facilities and the projects it supports.
Just to clarify the 111-marker vs 37-marker tests - you said:
"The next offer is Y-DNA111. I think I understand this to mean they will test more markers and give me a wider scope of tested relatives to confirm with."
Actually, it is the opposite.

It narrows the field of potential matches. That's what you want to do, as hurworth explained, since the fewer the markers, the more the "matches", although the matches are essentially meaningless at low numbers of markers, since those markers are common to so many people.
I found that it was next to impossible to understand what the heck it was all about until I had my male rellies' YDNA results in hand and had something to actually work with (a visual-kinetic learner!), and also great help from project administrators at FTDNA.
I'd dive in with the Y-DNA111 test and see what turns up - if anything, because keep in mind, it won't tell you anything unless you have someone to compare with: a match with a known or unknown person, or a mismatch with a known person, etc.