I'm not sure if this is a myth, but is it bad for a pianist to work out (e.g. lift weights)? I've heard that building strength this way will reduce fine muscle control.
This is a myth. An untrained person with large muscles will be just as clumsy as an untrained person with wimpy muscles. Fine motor control is a skill learned independent from the rest of the muscles in your body - if it were related, all brass players would have gi-normous lips and pianists would have gi-normous hands, and as well all know, this is simply not the case.
Physique only becomes a factor when a given body type will not allow a person to play an instrument with correct technique. For example, someone with a severe underbite who wants to learn to play clarinet. Other examples would be a person with fantastically large hands who is interested in oboe or a person with a weak jaw that wants to play tuba.
With respect to piano, it usually comes down to hand size - the larger the hands the easier it is to play. Certainly if you're of the muscular persuasion, piano will not hinder in the slightest as playing to the extremes of the instrument may be thought of like doing "flys."
Let me add some perspective here from a completely different angle: climbing.
Climbing is obviously a sport where people train a lot, and people "working out" in the sense of "sculpting an impressive body" tend to be amusingly bad (from the perspective of climbers not overly impressive to chicks while they keep their shirt on) at it.
That's because "working out" is generally geared towards gaining muscle mass. In order to achieve that, repetitions in training sets are geared towards the 7-15 rep range. Decidedly fewer reps (often as "excentric" loads somewhat akin to controlled yielding) only exercise maximum activation of muscle fibers, decidedly more reps are geared towards endurance, creating additional capillars and veins and circulation over time.
But the middle range used for body building profits from encouraging local buildup of storage space for nutrients and electrolytes around the muscle fibers doing the work but affects the actual available momentous maximum strength at best secondarily.
Since weight is a number one concern for climbers, and since the difficulties in a hard climb are either isolated (requiring quite brief peak bursts of power) or continuous for a longer passage (overhangs etc requiring endurance), the impressive "work out" musculature is dead weight not helpful for carrying you across larger passages.
For most music instruments you have a similar problem: of course you need bodily fitness and endurance and can profit from it. But what you don't need is having to move bulky muscle pads along with your fingers that have more than desirable inertia, compete for room to move with other bulky muscles, and that tire easily nevertheless.
So you should really first figure out just what you mean by "working out" and what goals your working out is supposed to achieve, and then figure out whether the goals in itself are incompatible in some manner with doing music fast and effortlessly and/or whether some particular manner of pursuing those goals might interfere more with your musical interests than necessary.
If you are aiming to turn your body into a coherent sculpture, spindly underarms make for a somewhat incoherent picture. But bulging underarms will interfere with what you can achieve in piano play.
I play the piano for more than 30 years. The type of workout that have been doing recently (past months) involves several different exercises using kettlebell weights. I am 100% sure that, if I play right after a workout woth kettlebells, my fingers do not respond to playing the piano in the usual way, they are more lazy, more stiff, it's like stepping back 15 years in my piano technique. The next day seems to be fine. I haven't figured out yet how the curve of recovery works in detail.