It is essentially matter of editorial style, not typography. The general rule is, as the notes constitute a sequence, both they and the note calls in the text stream should all be the same. Which is not to say that some editorial house style couldn't be different. IIRC, Chicago uses the same font & size for all note calls, which would be reflected in the their Manual of Style.

Having said that, good editorial practice usually moves a note call away from a subhead, and a chapter opening note is usually recast as an unnumbered note. A note to a subhead can also be so recast, esp. if the note is to the entire section the subhead introduces. But it is not the designer or compositor's job to make such changes.

Edit: It is possible for there to be two or more note streams within a text. If the text itself constitutes two streams, then a separate style could be warranted. For example, notes to tables are a separate stream, and have a different numbering & usually a different size. I suppose a text with a lot of sidebars might use a similar structure.

Alternatively, I have set text where there are two note streams throughout -- one the original author's notes, the second an editor's notes, as in, say, 17th century text, where the author's footnotes are preserved, but there are also editorial notes -- e.g., about what something might have meant at the time, or notes on differences in different editions. In books like that, the author's notes will typically be 1, 2, 3, etc., sometimes starting over each page, sometimes running as a sequence for an entire chapter. The editorial notes are almost always asterisk, dagger, double-dagger etc., and the sequence starts over each page. Typically, all notes are in the same font (includes size), but a designer could change this after conferring with the editors.

Worst I've ever handled had three sequences, one a commentary on a text with commentary on the commentary, the other a simultaneous tri-lingual piece.