Cohort commander?

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Roman epigraphs are sometimes confusing, especially if the space on the tombstone is rather small.
Tribunus Militum and Tribunus Legionis could mean the same. It is a Tribune of a legion.
Just Tribunus militum could also mean a praetorian tribune, if clear looking the context.
Tribunus cohortis always means a leader of an auxilia cohors.

Tribunus militum as a praetorian commander: do you have a reference for this? I'd expect tribunus cohortis x praetoriae.

In general, cohorts of Roman citizens had tribunes in command.

In the Republic, it was not unknown for Roman citizens to be levied into cohorts, rather than into legions - it happens during the Social war, and at various times in the civil wars. Either there wasn't enough time to marshall them into legions, or there weren't enough of them to form a legion, or both. It's the 'raise as many men as you can and bundle them into units' approach to military organisation, and effective in the short term.

This continued into the Triumviral/Augustan period. Some of these cohorts (Cohors VI uoluntarionum, etc.) were still commanded by tribunes (Purtisius Atinas, ILJug. 2) - although others, such as the cohors Apula (C. Fabricius Tuscus, AE 1973, 501), were commanded by prefects.

Why auxiliary units of Roman citizens would be commanded by tribunes, I'm not entirely sure, but I can make a guess. A tribune isn't just a commander. Just as the tribuni plebis had a responsibility to look out for the welfare of the people, so a military tribune had a responsibility to look out for the welfare of the troops, and to represent the views of their troops to the (pretty much all powerful) commander. Of course, all good officers make sure their troops are in good condition, morale, etc., but the title of tribune (and, I think, its historical function) conveys a sense that there was a formal duty of care that went beyond this. Tribunes of the people became more politically radical as time went on, and tribunes of the soldiers more like conventional military officers.

My guess is that it was thought appropriate for a Roman citizen soldier to have a tribune as an officer.

In addition, more prosaically, it also meant that even though these soldiers weren't in 'proper' legions, they still had the 'right' officers.

However, there was a ranking amongst tribunes depending on the rank of the unit the tribune leaded. A tribune of the praetorians or of an ala millaria was higher ranked than a tribunus cohortis or tribunus legionis. Also in late empire there were multiple ranks of tribunes.

Regarding the equestrian career there are 4 ranks (until the Flavians just the lower 3)
1. praefectus cohortis (quingenaria)
2. tribunus militum legionis or tribunus cohortis (millaria)
3. praefectus alae (quingenaria)
4. tribunus alae (millaria)

Minor quibble: you can't really generalise about the pre-Flavian equestrian cursus, it's all very fluid. In the Augustan period, it's incredibly fluid, and infantry cohort commanders are fairly rare. So, from Augustus to Tiberius, it's generally:

1. command of an auxiliary infantry cohort (optional), either as praefectus or tribunus, depending on the unit
2. legionary military tribunate
3. command of an auxiliary equestrian unit

with other commands as necessary coming afterwards. However, centurions can be promoted directly to cavalry commands, and most officers only hold one command.

By Claudius, command of an auxiliary infantry cohort is more common, and there are more of them. But Claudius, famously, reverses the order of the other two posts:

1. command of an auxiliary infantry cohort
2. command of an auxiliary equestrian unit
3. legionary military tribunate

It's only really under Nero that the classic pattern starts to settle down, partly as the Roman army starts to settle down. Rome doesn't really have a single 'army' in terms of higher-level command structures, organisation, and management until the Augustan period. It's no surprise that it takes 50 years or so for the command structures to stabilise.