Well, I would say you're both right, but each focused on only part of the issue.
For safe, reliable charging of lead acid batteries, you need to control, or at least limit, both the current and the voltage.
The current needs to be limited to the lower of either the safe charge rate of the battery, or the safe output power capacity of the charger.
The voltage needs to be limited to prevent overcharging of the battery. That level varies somewhat depending on usage and temperature. Here, Mike's link is a great source of info.
Please note: Both of these limits are mainly to prevent bad things from happening, to either the battery or the charger. The actual mechanism of charging a lead acid battery, is simply pushing properly polarized current through it. The voltage point is when to stop, or significantly reduce, the current.
The current does not need to be "constant", and many chargers aren't. See "Taper chargers" in Mikes link.