What unit (TPM, FPKM/RPKM, or other) to use when working across samples
Neither of RPKM, FPKM or TPM is a good choice. All these methods are similar, they only perform the operations in a slightly different order. Eventually they correct for gene length and sequencing depth. Correction for sequencing depth is necessary but not sufficient in many cases. Correction for length is typically not done for a differential analysis as it reduces the counts of many genes and therefore throws away statistical power. I will not describe this in detail since the StatQuest video series you link includes videos about DESeq2 and edgeR normalization procedures which extensively cover the normalization procedures and why these approaches are superior to RPKM/FPKM/TPM. In short: You have to correct for differences in library composition to compare between samples. Please watch these videos and read the manuals of the established RNA-seq analysis packages such as but not exclusively limma, edgeR and DESeq2.