The name baby Botox sounds like a product, but it is really a claim about dose. The first job is to find out who defined the phrase and what they say it means.
The definition on record
Cleveland Clinic explains it this way: "This cutesy name refers to the fact that it uses smaller doses of botulinum toxin during the procedure — that is, “baby” amounts." So the name describes an amount. It does not name a different drug, and it does not tell you how much anyone will actually inject.
Reference pages versus the nickname
For general background, the MedlinePlus page on Botox and the DermNet topic on botulinum toxin cover the product itself. Their titles do not mention the nickname. Check whether either page uses the phrase before you treat it as established terminology, because a name can spread faster than its evidence.
What "smaller" assumes
A small dose only means something next to a standard dose. A proposal that says "baby" without naming the comparison leaves you unable to judge it. The word assumes you already know the usual amount, and most readers have no reason to.
Smaller than what?
Find out what the smaller dose is being measured against, and which areas of the face it is meant for. Both deserve a plain answer, and hearing the nickname again with a smile is not an answer.
Where the product family fits
Cleveland Clinic's page on neuromodulators, including Botox, Dysport and Xeomin groups several products together in its title. If a quote names only one brand, ask whether the provider considered the others and why this one was chosen.
Separating the dose claim from the timing claim
Keep the nickname separate from the timing question. Cleveland Clinic asks a different thing in Does Preventative Botox Really Work?, and a quote that blends both ideas deserves to be split in two. The practical comparison is simple, and it is the same for every reader this publication serves, from its North Fresno base out into Madera County: write down the dose claim, the product named and the timing claim, then check each against a source that has no stake in the sale.