… if I cast Booming Blade, am I locked out of Extra Attack because I took a “Cast a Spell” action instead of an “Attack” action?

Yes. Extra Attack specifies that you must take the Attack action. With Booming Blade, you’re still taking the Cast a Spell action, the singular melee weapon attack is simply how the spell is cast, not an entire Attack action.

This is the same for all spells that require you to make a weapon attack as part of casting it. The only spells that kind of break this rule are the various “Smite” spells, but they’re worded like “When you next hit a creature with a melee weapon attack…”, and generally use a bonus action rather than an action, so you can still use your action for the Attack action.

The action you're taking is Casting a Spell.

Attack is a different action which would let you attack twice with Extra Attack.

You cannot Booming Blade your attacks with Extra Attack.

This is actually the topic of a Sage Advice:

Can you use green-flame blade and booming blade with Extra Attack, opportunity attacks, Sneak Attack, and other weapon attack options? Introduced in the Sword Coast Adventurer’s Guide, the green-flame blade and booming blade spells pose a number of questions, because they each do something unusual: require you to make a melee attack with a weapon as part of the spell’s casting.

First, each of these spells involves a normal melee weapon attack, not a spell attack, so you use whatever ability modifier you normally use with the weapon. (A spell tells you if it includes a spell attack, and neither of these spells do.) For example, if you use a longsword with green-flame blade, you use your Strength modifier for the weapon’s attack and damage rolls.

Second, neither green-flame blade nor booming blade works with Extra Attack or any other feature that requires the Attack action. Like other spells, these cantrips require the Cast a Spell action, not the Attack action, and they can’t be used to make an opportunity attack, unless a special feature allows you to do so.

Third, these weapon attacks work with Sneak Attack if they fulfill the normal requirements for that feature. For example, if you have the Sneak Attack feature and cast green- flame blade with a finesse weapon, you can deal Sneak Attack damage to the target of the weapon attack if you have advantage on the attack roll and hit.

Also, you actually can use Booming Blade twice in one turn because its a cantrip. The limit for spells cast only applies to leveled spells. The easiest way to Booming Blade twice is the Sorcerer metamagic Quicken Spell.

As a caveat to what has been stated by others, if you're a bladesinger wizard, the Extra Attack there does allow for it to be used with Extra Attack since Bladesinger allows one of the attacks to be replaced with a cantrip cast.

Others have already made the point that attacking is not the same as the attack action. The consolation is that the extra damage from Booming/Green-Flame Blade goes up at level 5 instead!