If you test with 23andMe, you will get haplogroup information (mtDNA for males and females; Y-DNA for males). These haplogroups are an estimate as they rely on a small number of SNPs to determine your haplogroup. In most cases, you will get a basic to intermediate haplogroup result using 23andMe or any other company or website that analyzes your DNA using SNPs. LivingDNA, and by extension – FindMyPastDNA, use SNPs.
If you want a full haplogroup, then you need to either do a full DNA test or use a company that offers full mtDNA testing and/or deep Y-DNA testing. FamilyTree DNA (FTDNA) offers full mtDNA testing and fairly deep Y-DNA testing, but neither test is cheap. While FTDNA’s Big-Y 700 is one of the better Y-DNA tests on the market, it’s still not a full Y-DNA test.
Full Genomes Corporation’s (FGC) Y-DNA testing – https://www.fullgenomes.com/y-chromosome-sequencing/ is another great option, and in terms of pricing, it’s $425 compared to FTDNA‘s Big-Y 700 at $649. It was a more comprehensive test than FTDNA’s Big-Y 500 test although I haven’t seen comparisons between FGC’s Y Elite – https://isogg.org/wiki/Full_Genomes_Corporation and the updated FTDNA Big-Y 700. Although I expect the edge will still go to FGC. I expect somebody will do a comparison between Y Elite and Big-Y 700 in the near future.