They wouldn't have only a +1 weapon, they'd just get +1 free, and pay for whatever else they want above that.
No, they would get a +1 weapon, and if they wanted to have a +2 weapon, they would have to first pay for a +1 weapon, then pay for a +2, and would be exactly like a Barbarian 20 except have no Samurai class features at all. (Okay, the weapon would still be Ghost Touch.)
That's the point. A Samurai 1/Barbarian 19 is still a Samurai for the purpose of the level 1 class feature "Ancestral Weaponry" which is a Samurai class feature.
So when a class ability says "a Samurai of Xth level" it's referring to the fact that you have to be a Samurai to have that class feature.
Saying "Traditionally, when a class refers to a PC's level without clarifying for character level, it means class level." is begging the question. How do you know that's what it means in those situations, except that your made your judgement previously, and are now imposing it on every instance.
Frank doesn't have to go on record saying that it's character level for me to know that the RAI matches the RAW of character level, because he also made the Soulborn class with a class feature:
Which only scales differently at all (no cap, round up instead of down) because it uses the Book of Gears magic item scaling system as a base, which didn't exist when Samurai was written.
I know the guy well enough to know that he doesn't think you should arbitrarily punish multiclassing warriors, since he was posting warrior multiclasses as the only viable warrior builds back in 3.0.