IPhone – put the iPhone 4 sim card into a new iPhone 4S

No - the sim card only provides an ICCID number to the phone. That number is read once per boot as the phone just uses the sim to identify what account to bill for service when registering with the nearby cell towers (IMEI and ICCID together let the system know what number to assign your phone and who to bill).

Something else must be causing your observed behavior.

Keep in mind, there is no guarantee two phones in the same location are not on distinct cell towers. Especially if one tower is overloaded or doesn't have high speed data so the phone /carrier might pick a faster connection that happens to have less bars of signal.

What does your carrier say if you call them for a site survey / diagnostic?

There are two different iPhone 5s models sold in the US -- the CDMA model for Verizon or Sprint (which also supports GSM connectivity for international use) and the GSM model. The unlocked phone is the GSM model, so it won't support CDMA and won't talk to Verizon's network (except perhaps for data (not voice) on 4G/LTE, where Verizon uses GSM like the rest of the world). So you can buy an unlocked phone and put a Verizon SIM in it, but it will at best only work with Verizon for data, and only in areas where they have 4G coverage.

Just look at the bottom of the page you link to in your question, where it says:

If you buy a SIM-free iPhone, you will need to purchase a GSM-compatible nano-SIM card separately. In the United States, you can purchase such a SIM from AT&T or T-Mobile.