Wonderfully done! I’ve never listened to Elliott Smith, but your performance sounds incredibly professional to me. Yes, the tempos change from section to section, but if I hadn’t read your post, I honestly wouldn’t have realized that this wasn’t intentional.
If you want to steady things more tempo-wise, there isn’t much to do about this other than to practice more with a metronome. I am probably not the best person to give advice on this, as I don’t play with a metronome as often as I should myself. But I was an aspiring drummer at one point, so I’ll take my best stab anyway. My apologies if I cover something you already know! Just doing my best to sum up the topic as best I can.
Your experience that you play faster when excited is a very common one and is a big reason most musicians need metronome practice. If you find it too difficult to keep up with the metronome in some sections, definitely slow it down to a tempo where you can comfortably and confidently keep up with it, even if this means a painfully slow speed! You should never try to play something faster than you can really handle, and this is the main reason to play with a metronome rather than the recording. If you have one of those apps where you can vary the recording’s tempo, or if you can already comfortably play at the recording’s speed, then go ahead and do that if you find that more fun!
It’s okay for it to feel painfully slow. That’s just what happens when you’re not used to using a metronome. It may also be very frustrating to find cases where you struggle to play something slower than you are used to or are able to, but that, too, is a natural reaction, and a sign that this is a passage you need more work on. True mastery of a part means being able to play that part at a variety of speeds. Learning the places where you tend to rush and learning how to hold back will take practice, but once you’ve done that a few times, it should become easier. It may help to have the metronome go every eighth note instead of every quarter, or perhaps some other subdivision.
One great tip I learned from a Benny Greb video was to let your body move to the beat while you listen and while you play to help you keep time, whether it be bobbing your head, tapping your foot, swaying your whole body, whatever comes most natural for you without interfering with your playing/singing. I suggest doing this both with and without the metronome.
The transitions from section to section are probably going to be the hardest, and so you’ll want to concentrate your practice on those (i.e., start right before the transition and end right after, then repeat just that section until you can do it smoothly, then make sure you can still do the transition smoothly if you start a few more measures before that).
Personally, I think performances/recordings should always be without the metronome, and that the metronome is just a practice tool to help you get a better sense of steady time, but there are some professional performers who swear by a click when performing, so decide for yourself which performances you are happiest with.
Anyway, I hope this didn’t come across as a lecture or condescending. I think your performances were fantastic and probably much better than anything I’ve done. This is just a topic I’ve spent a lot of time researching.