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Impossible to say, could be the app is doing something funky, could be iOS, could be lotta things.

I will note, my preferred solution is to do none of the above, and I only do split DNS for one particular service. I much prefer just using an always on Wireguard VPN that is set to only route traffic to my internal subnets and to use my internal DNS server. Then I just use internal names. Wireguard basically runs at line rate on my setup, so half the time I don’t even turn it off at home. This also gives you the option to use DNS ad blocking (eg adguard) on the go.

Hmm, caching has never caused problems with split DNS for me, but it’s really hard to debug what was going on with your setup. Split DNS is really common and is the preferred way to solve this, so most browsers have logic to handle it. You might have had something misconfigured, but unfortunately it’s really hard to diagnose.

AKA, split DNS. Doing it this way is a bit cleaner than hairpin NAT as mentioned in other comments, but both options work fine in a home network.