2013 Elantra GT starting problem

Hi all -- i am having starting issues with a 2013 Elantra GT (manual transmission) that just crossed the 100000 mile mark.

So i was driving from austin to dallas and stopped at a shell about an hour out of Dallas. When i tried to start it all i heard was a loud clicking sound at regular intervals as long as i pressed the clutch. I thought that the battery might be the problem. Luckily i had a jump starter with me. Connected it, gave it a try and it started without any hiccups. So now i was sure it was the battery. Used a friend's car while in dallas and simply jump started mine on my way back to Austin and didn't switch it off the whole way.

Next day i called AAA and asked them to come change the battery. For some reason that day the car was starting without having to jump start it. The guy connected a device to the battery terminal and announced that the battery is 100% fine, the starter test is also fine but the alternator is not working. I asked him to repeat the measurement and got the same result.

Next i took it to a O'reilly and asked them to check the battery for me. They did and said it was perfectly fine. The guy asked me to start the car so that he could test the alternator but the car suddenly decided to not start and repeated the ticking/clicking sound. The guy said that sound is unmistakably the starter motor. It's not engaging. I then asked him that if that's so then how come jump starting it works. His answer was that there is some life left in the starter and jump starting it just provides the extra push. You agree? He got his own jump starter but this time the car refused to start with that as well. Luckily on the third try it did.

Now i was confused between the alternator and the starter. I ruled the battery out because two of those fancy machines had ruled it out. Saw a bunch of videos on starter/alternator issues. Looks like the car people don't like these fancy devices and prefer a simple multimeter. Got one. Here's where i am at now. I measured the battery voltage with the car off. It's 12.6 V. Started it by jump starting it, the voltage at the terminals is about 14.4V which suggests the alternator is ok. Switched on the lights, vipers etc and then the voltage drops to about 14.3 V. Did a voltage drop test from the negative terminal of the battery to the alternator body. Nothing there to report. Here's what i the diagnostic procedure looks like in my head:

-- all the relays/fuses involved in the starting procedure are fine because jump starting it has been working more or less (i also started it by driving it down a small slope once)
-- the terminals on the battery seem solid and clean
-- the alternator is also fine because if it was not then the engine would have stalled on the way from Dallas to Austin as the battery can't support it for 4 hours. Also the multimeter readings are good
-- i want to get to the starter to see if it's getting consistent voltage and to rule out wiring problems. The problem here is that i don't have car jack stands or ramps. Is it a good idea to invest in those?

Assuming the voltage is consistent at the starter then i guess the simple explanation is that it's time to change the starter. Should i get a new one or get this one repaired? Also is it a good idea to invest in (car jacks from earlier) and a tool set to try and replace it myself or should i just get a new starter once i have ruled out wiring problems and take it to some mechanic and ask him to simply change it.

And just for completion, my latest jump start today somehow lit up the check engine light. But the car drove fine. So maybe it's just something temporary and will go away tomorrow. Also the first time this happened on my way to Dallas, i noticed that the speedometer and the RPM meter were behaving weird. The needles were sort of moving slightly and the lights dimming. Haven't noticed this since.

Any ideas/mistakes in my diagnostics/advice would be very much appreciated by this newbie . Thank you reading this long post.