Why is lutetium part of the lanthanide series?
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.