Why is lutetium part of the lanthanide series? I’ve been looking at the periodic table and began to wonder - why is lutetium part of the lanthanide series? Or, for that matter, why is lawrencium part of the actinide series?

This isn’t readily apparent when looking at the traditional periodic table because the lanthanides and actinides are separate and below the main table. However, on the extended table, the two elements don’t appear that they should be in the groups that they are in: lutetium contains the $5d$ electron shell whereas the lanthanides do not, and lawrencium contains the $6d$ electron shell whereas the actinides do not.
The assignment of elements to groups or series in the periodic table is not based exclusively on the electron configurations concerned. Helium for example is an s-block element yet is placed above Ne in the p-block due its noble gas properties.

Lutetium is part of the lanthanide series, notwithstanding its electron configuration, because its properties and those of its compounds are similar to those of the other lanthanide elements.

Lawrencium is assigned to the actinides for basically the same reason. Experiments have so far failed to find any chemistry-based evidence for Lr having a p electron. While it may have a p electron in the gas phase it probably has a d electron in the condensed phase.

Whether either Lu and Lr, or La and Ac are assigned to group 3 will not make any difference to the composition of the lanthanides or actinides, since both of these are horizontal series rather than vertical groups.