I enjoyed Lillie's character development during the first year of SM and during the Aether arc and I enjoyed the episode where she battled her Butler and the Tyranitar.
To pinpoint one myself, I'd say the 'Effortful Lillie'/'Family Determination' episode is Lillie at her fullest. It's practically a character study for her, demonstrating nearly every key personality trait she has, which isn't a small package (probably why it feels a tad compressed).
* She's very neurotic, being meek and awkward (early on when Gladion ruins her poignant image change).
* She is not self confident, though tries to fake it with varying results.
* She is none the less assertive about getting her voice heard and sticking to a goal.
* She is empathetic towards others' flaws in spite of this, just let her get to understand them enough (when Ash puts his foot in his mouth on the ride home, Lillie quickly breaks from her mournful attitude to smile and assure him it's fine, she knows it's just Ash and he means well).
* In the Japanese version at least she is very formal and polite.
* She prefers a studious approach, but can make impulsive choices in a pressured situation.
* She is intellectual, but in a particular element. In other areas she is a niave buffoon (she has a moronically romanticised view of Poni ruins due to having 'read about these things in a book').
* She is sheltered, she isn't used to epic physical adventures, though will always try her best in spite of her nerves and clumsiness.
* She is not a strong trainer but she is a smart one.
* She is very affectionate. She loves her Pokemon in a gentle way, is loyal to her family in spite of arguments (beaming just from passive approval from her brother) and while trying to do things herself can be quickly convinced to accept her friends' help.
I love how the whole episode is a deconstruction of the 'determinator' trope as well. In most cases in the anime (and most other media for that matter) the "getting determined" moment is generally the key to winning, once the character has that epiphany they become unstoppable. Lillie...isn't quite as successful, there's clearly a lot more to it in her situation. She is the embodiment of a 'try hard'. I love how we see how so different yet so similar she is to Ash in this episode, being good in her area of expertise but screwing up and acting like a comical butt monkey out of it. It may not perfectly fit the serious stakes, but it sure gives Lillie tons of pathos, especially since the humiliation and feeling of being worthless makes her less capable of bouncing back like Ash.
There is a battle, and while it's not the most tense, it bothers to show Lillie's niche in the field a lot better than usual unofficial battles, which tend to just be quick by-the-numbers thrashings with little substance. Here however Lillie is in a vindication climax, stumbling for the first few moments but then using her strategic strengths to help Gladion when they lose the upper hand.
Despite his limited role I think this is a good moment for Ash as well. This is a very alien position for Ash at the start, he's within the tragic aftermath of a disaster happening, it being unknown if it can be solved and even being asked to not get involved and do his usual save the day routine. He is clearly not used at all to standing by and dealing with the heavier elements like these. He tries to put focus on caring for the Pokemon like he always does, but quickly realises this is completely inappropriate for the time (see above). However when he realises Lillie has gone into the thick of things alone, he gets back into his comfort zone instantly, leading the front. Even better since given SM's standstill premise, Ash has to actually form a route himself, making him feel more proactive, he seeks advise from Hala and makes a beeline to another island in the region. The fact he manages to so quickly catch up with the two in time to save them from debris falling is questionable, but it fits Ash's character here. Altruism is always his forte, and makes clear to Gladion he's helping whether he wants him to or not.