Two legal entities doesn't mean they're washing their corporate name, both are still called Motorola. It's completely normal to have subsidiaries for countries you have employees or do a lot of business in, it greatly simplifies accounting. It's even normal to have subsidiaries for your different business activities. I suspect Motorola is actually quite proud of their work in Israel, it's an American company after all.

Buying a phone from Motorola Mobility, a completely unrelated company owned by the Chinese spyware company Lenovo, gives money to neither of those companies.

Instead, you'd want to avoid using telecoms that use Motorola tech. In North America that includes Verizon, AT&T, Bell, Rogers and Telus, possibly others. Europe has too many to even find.