Infinitives with causitive verbs bid, command, order

You are confusing "meaning" with "grammar". As you correctly observe "bid" and "order" have a similar meaning in this context, but different grammar. There is no strong reason why.

Note that this sense of bid is now dated, or even archaic, and so shouldn't be used except in fixed expressions like "bade farewell". There is a relatively small list of verbs that take a bare infinitive: "bid, dare, make, see, hear, let" and even among these, a to-infinitive is sometimes possible (dared to go, but durst go) In the past there were more verbs like this, now the bare infinitive only holds on in some very common verbs (like made) and a few archaic verbs (like bade)