I had one middle school teacher call them wuzzles (word puzzles) and I’ve seen them called webus (because a rebus is pictures so a webus uses words.).
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Definitely Dingbats - the Mail on Sunday (UK newspaper) has them on their puzzle page every week. Though quite possibly “Dingbats” is a trademark as well - I’m pretty sure there’s a board game version.
ETA: I feel I must point out that I rarely read (much less buy) the Mail on Sunday :).