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You are correct. Most of us are "on a budget" when we travel in the sense that we have certain price ranges in mind even if we tend to prioritize saving time over $$ and stay in fairly nice places. Budget travel or traveling on a budget carries the implication that you're looking for cheaper options like buses, hostels, free attractions, no-frills airlines, etc. Think the original Lonely Planet demographic (though not any longer).

And with an upper limit on costs for Hotels and Flights.

If I may make a comparison, kinda like a half-baked Expedia without the $£€ limit filter.

Native English speaker - that was my original expectation as well. It can mean either thing.