As indicated above, the single most important property of a vacuum pump oil is not so much its lubricating ability, but rather its vapor pressure.
Essentially, in a rotary vane pump, even though the oil DOES lubricate the mechanics, its more significant function is to both seal the vanes against the housing AND also actually physically transplant the pumped gases out of the pump. You can observe the latter directly when you first power on a pump and it's either pulling down the vacuum system from atmospheric pressure, or if there's a massive leak-the pump will make a "gurgling" sound, usually accompanied by either mists of oil or sometimes larger droplets "burping" out the exhaust. The theoretical minimum vacuum a pump can achieve is limited by the vapor pressure of the oil. Sometimes in research labs, pumps get used(abused) to pull off solvents without adequate vapor traps, and also don't get oil changes as often as possible-that leads to the oil loading up with solvent and increasing the vapor pressure. I recall working in one lab where an old Sargent-Welch belt-driven pump was used several hours a day as part of a set-up that removed carbon disulfide(too lazy to look it up, but I think it boils at around 45ºC) and no one could remember when the oil had last been changed. With a gauge directly hooked up to the inlet, I measured ~100 torr, which is miserable for this style pump. The oil I drained out was a lot less viscous than what I'd expect pump oil to be(plus it smelled like CS2), combined with a lot of sludge that came out at the end. We had a 55 gallon drum of "#19 pump oil"(generic branded Inland 19) out back, so I filled a bottle and ended up changing the oil 3 times that day. The end result was a pump that was not only quieter, but would reliably pull 10^-2 torr. Unfortunately, I was only there for 3 months about 10 years ago, and it wouldn't surprise me if the oil hasn't been changed since...(although the pump might have died without that).
In any case, in all that rambling, unless I have a good reason to use otherwise, Inland #19 is my go-to, and it wouldn't surprise me if the linked NAPA oil is a generic of that. It's basically just a heavy hydrocarbon oil.
https://www.capvac.com/inland-19/
Inland 45 is my preferred oil for things like mass spec use, although it's ~4x the cost of #19 and for HVAC work I seriously doubt you'd see the advantage of it. I've used Inland 19 for mass spec applications in a pinch, and it's been perfectly serviceable(in fact I have one running it now-changing it is on my to-do list for the week). The only real failing is I see a somewhat higher hydrocarbon background than Inland 45, but then I can actually directly measure that(and identify that as the issue)-I doubt you'd see it with an HVAC system.