So. I’ll answer your TLDR question first, but I think there’s also a couple of things to comment about your mercenary group.
In short? However you want them to, but nothing is official.
The only actual common meaningful military titles in the Empire are General (“person enshrined with authority to command an army”) and Adjutant (“person appointed as General’s second”). Different player groups and nations might use other titles like Captain, but ultimately they only mean what that group/nation broadly agrees they mean.
As @thresher says, though, none of these ranks necessarily “mean” anything in the game itself. Apart from a General being able to control an army (which never includes the actions of an actual Player Character), there is no ability to order anyone to do anything.
So the Generals & Adjutants (plus some ancillary titles) report to the Military Council (which is formed of all of them), which has its own couple of ways of feeding through to the Senate. Incidentally, it’s the Senators of each nation who select the Generals.
On the field, there are concepts such as the Field Marshal - someone who the Military Council appoints to be the overall leader of the Imperial forces at an uptime battle - who reports back to the Military Council. But again, they have no legal power to order anyone to do anything (which doesn’t mean people won’t listen to them).
Off the field, it can be presumed there are all sorts of different positions within each Imperial Army. I know Generals have received IC messages from people in their Armies - I think Plot Writers have just looked around for a title that “felt right” rather than consulting an actual list, because it’s likely both beneath the abstraction layer (i.e. too much detail to worry about specifying as a Game Concept) and different for each army.
The Civil Service also collates reports on all the Armies - presumably through some sort of bureaucracy or hierarchy within said Armies combined with divinatory magic - and presents this to the Military Council each summit. Again the details of this are left below the abstraction layer unless they need to be specified for a particular Plot.
Cross-nation groups are explicitly discouraged by the game. You can’t be part of the same band mystically, which means you can’t benefit from group magic items, and a lot of your means of co-operation are taken away. It’s possible to have a Sodality - a cross-nation organisation of some kind - but they tend to be more than just a band of fighters.
Also, national politics and plot will likely take people towards their nation rather than the group you’re making. Have you considered possibly more discussions and a potential compromise on nation choice if you do want to actively play together?
A Mercenary Group is quintessentially League (a Free company), but it’s not a very Wintermark concept. Wintermark is riddled with the concept of heroism, whilst Free companies are typically driven by profit.
And as others have said, there may well be issues with fighting together - if, say, Wintermark and the League are fighting different battles (one on Saturday and one on Sunday), then you won’t be able to take the field together without getting a Mercenary Banner just for one of you. You might be able to find some skirmishes (which generally aren’t nation-limited), but they tend to be targeted at specific nations.
Having a League free company with key ideas about fighting together is a great concept. Mercenary groups tend to find it a little tricky at Empire - where you could theoretically charge people for the pleasure of going on skirmishes rather than paying people to risk their lives - but it has been done and people get a lot of enjoyment out of it.
I guess what I would ask back is: why do you want to play in the League? Why does your friend want to be a Winterfolk? What other nations do your friends want to play, and why? You might be able to find some common ground there.