i'm sure high end solnedoids are quiet.
every one i've ever encountered (for pop locks, nitrous bottles, header dumps, flames out the tail pipe trick, and more) all make noise... in extreme applications & when testing at idle or with the engine off ...to say the least.
that timer could be noisey too.
i've only seen those timers, never worked with one.
best to go over all wiring to be sure.
i didn't check those links yet. if i need to, let me know
when i'm faced with a situation like yours, and as long as i will not ruin the device (cause there's a fuse, shut off, etc), and if there is another VALID wiring option... i'll always try that option.
i only do the above after the result of testing and deduction WITH a meter.
you already have several "knowns" which is quite excellent for a reh-tard if i may say so.
my word of advice, & cause your hobby demands it, plus you can use it on your car & electrical devices, get a decent ($20 to $50) multi meter.
believe it or not, i've never used one of those pens.