Hi David,
Regarding the difference between concentration/tranquility/samatha on one hand, and noting/vipassana/investigation on the other, I'd suggest a little experiment:
To get into tranquility, it really helps to relax the body, so do the "relax on in-breath, relax on out-breath". To relax, imagine your favorite bubble bath, or remember how it feels to sink into bed after a hard day's work, or imagine your muscles being warm and heavy. You can combine this with verbal "in" and "out" notes, or a meditation word. You can also do something like PMR (progressive muscular relaxation - google the term, there are many online instructions, and I think it was discussed here recently) before meditating.
To get into noting/investigation, instead of an altered state, closely feel "into" the breath sensation, (whether at nostril or abdomen or wherever - chest is also interesting, as is back of throat, and so on), and then make a note whenever you notice any change in that sensation, such as it becoming faster, slower, cooler, whatever. You want to "slice" it into thin samples, instead of smoothing it down with the meditation word like you did in the above exercise. You can use the same notes, "in" and "out", but use them to note each time you notice you're still breathing in or out, which will be multiple times per in- or out-breath. Go for speed over accuracy, but don't get into a trance of quick automatic notes - that would be again a tranquility practice. You want to notice the sensation, then acknowledge that with a verbal note.
Experiment a bit, and you'll see what is meant by the difference from your own experience.
A nice analogy (if you like analogies) by Kenneth Folk is: imagine you have a pillow case. In Samatha, you don't care what's inside, you put your hand onto the smooth, soft pillow. In Vipassana, you want to put your hand into the pillow case and find out what it is filled with.
I view the vipassana/samatha distinction as a spectrum, with samatha at one end, and vipassana at the other, and many "mixed" practices in between. If you enjoy sifting through archives, there are many old discussions here exploring the difference.
Finally, many people find kasina practice - gazing at a dot or colored disc or candle flame - to be initially easier to develop fixed concentration than the breath, so you might want to give that a try. Oh, and getting stream entry definitely gives concentration a big boost - something to consider even if that's not foremost on your agenda for some reason

Cheers,
Florian